AA GENERALIZED FIBONACCI SPIRAL
BERNHARD R. PARODI
Abstract.
As a generalization of planar Fibonacci spirals that are based on the recurrencerelation F n = F n − + F n − , we draw assembled spirals stemming from analytic solutions of therecurrence relation G n = a G n − + b G n − + c d n , with positive real initial values G and G and coefficients a , b , c , and d . The principal coordinates given in closed-form correspond tofinite sums of alternating even- or alternating odd-indexed terms G n . For rectangular spiralsmade of straight line segments (a.k.a. spirangles), the even-indexed and the odd-indexed di-rectional corner points asymptotically lie on mutually orthogonal oblique lines. We calculatethe points of intersection and show them in the case of inwinding spirals to coincide with thepoint of convergence. In the case of outwinding spirals, an n -dependent quadruple of pointsof intersection may form. For arched spirals, interpolation between principal coordinates isperformed by means of arcs of quarter-ellipses. A three-dimensional representation is exhib-ited, too. The continuation of the discrete sequence { G n } to the complex-valued function G ( t )with real argument t ∈ R , exhibiting spiral graphs and oscillating curves in the Gaussian plane,subsumes the values G n for t ∈ N as the zeros. Besides, we provide a matrix representationof G n in terms of transformed Horadam numbers, retrieve the Shannon product differenceidentity as applied to G n , and suggest a substitution method for finding a variety of otheridentities and summations related to G n . Introduction
Spiral representations of the (integer valued) Fibonacci numbers and of some of their gener-alizations (e.g., [43]) are known for long and applied both naturally and artificially in two (e.g.,[7], [16], [15], [9], [32], [25]) as well as in three dimensions (e.g., [14],[30]). For a non-exhaustingreview on applications of the Fibonacci numbers with some more occurences concerning spirals,see [35]. The underlying number generation principle is equivalent to a deterministic nonho-mogeneous linear second-order recurrence relation as treated in most introductory textbookson difference equations (e.g., [23] or recently [8]). Investigations with the genuine Fibonaccisequence and with some of its various generalizations are often involving an explicit Moivre-Binet-form solution. This is particularly encountered with Horadam sequences without [2],[18] and with an additive constant input [1], [41], [12], [34]. The explicit solution for the gen-eralized case with real constant coefficients and with an additive exponential input has beenrecently discussed by Phadte & Pethe (2013) in [33] and Phadte & Valaulikar (2016) in [34].Here we extend their work, firstly, by a couple of finite sums that are applied to, secondly, avisualization of such generalized Fibonacci numbers by means of assembled spirals. Similarto the generation of the original Fibonacci spiral, our piecewise construction uses parametricplots for the interpolation between principal coordinates of each segment. We focus on rect-angular spirals (a.k.a. spirangles) and on arched spirals, both in two and three dimensions.
Date : 20 April 2020.
Key words and phrases.
Linear second-order difference equations, nonhomogenous recurrence relations, ex-ponential inhomogeneity, generalized bivariate Fibonacci sequence, transformed Horadam sequence, Shannonidentity, Fibonacci curves and spirals.... a r X i v : . [ m a t h . HO ] A p r ur graphical representations are restricted to lines, we won’t draw corresponding spiralingsurfaces in 3D space here. Similar to the treatment of Horadam & Shannon (1988) [21] andHoradam (1988) [22] for Fibonacci, Lucas, Pell, and Jacobsthal curves, or recently by, e.g.,Chandra & Weisstein [6] and ¨Ozvatan & Pashaev (2017) [32] for Fibonacci numbers, we ex-hibit the exponentially generalized Fibonacci sequence by means of a continuous parametricplot in the complex plane. This produces, dependent on the parameter values chosen, eitheroscillatory or spiraling graphs.The content of the paper is as follows. In Section 2 the explicit (non-degenerate) solutionfor the generalized Fibonacci sequence with exponential input is derived, a related generalizedproduct difference identity of order is formulated and shown to correspond to a generalizedShannon identity, and the finite sums of alternating even- or odd-indexed terms are given inclosed form for proper use in the sequel. In Section 3, following a short ad-hoc classificationof spirals and relying on the results of the previous section, the formalism proposed to drawrectangular and arched two- and three-dimensional generalized Fibonacci spirals is presentedand graphical examples are shown. Specific features like the orthogonal positioning of direc-tional corner points in outwinding spirals or some point of convergence for inwinding spiralsare discussed and calculated. In Section 4 the index n is replaced by some real variable t ,leading to generalized Fibonacci spirals and curves in the Gaussian plane. The conclusionsare drawn in Section 5, followed by the Appendix in Section 6. The latter harbours a matrixrepresentation for the generalized Fibonacci sequence and a related decomposition into funda-mental Horadam numbers as well as some summation formulae, including the principal proofof the paper.2. Generalized Fibonacci sequence with an exponential input
The deterministic nonhomogenous linear second-order difference equation G n = a G n − + b G n − + c d n , n ≥ G , G ∈ R and subject to real constant coefficients a , b for the homo-geneous part and alike c , d for the inhomogeneous input, has a well-known solution, as recastbelow for the sake of completeness. If the inhomogeneous input is written as c d n − m , m ∈ Z ,all the following results stay valid with a correspondingly substituted value c d − m instead ofour choice c . Actually, in [33] and [34] the value m = 2 is used. The sequence corresponds toa generalized Horadam sequence [18],[27], with the generalization being due to the exponentialinput and due to the allowance of real parameter values (see below). In the same sense, onemay call it a generalized bivariate Fibonacci sequence or a generalized Fibonacci polynomial in a and b (e.g., [3]) with an exponential input . For the sake of simplicity, we restrict our inquiryhereafter to parameter values a , b ∈ R + and c , d ∈ R +0 .The formal extension to negative indices − n < G − n +2 = aG − n +1 + bG − n + cd − n +2 , yielding G − n = 1 b (cid:110) G − n +2 − aG − n +1 − cd − n +2 (cid:111) . (2.2)In particular, applying this twice provides for later use G − = 1 b (cid:110) G − a G − c d (cid:111) (2.3) G − = 1 b (cid:110) ( a + b ) G − a G + c ( a d − b ) (cid:111) . (2.4)2ENERALIZED FIBONACCI SPIRALFor practical purposes, any negative integer index value may simply be plug into the formulafor the general solution that will be derived in the next section.2.1. Explicit solution.
Inserting the Ansatz G n = A λ n into the corresponding homogeneous(or reduced) recurrence relation (2.1) (i.e., with c = 0) yields the characteristic equation λ − aλ − b = 0 , (2.5)with the fundamental set of solutions α := λ , β := λ given by the quadratic formula, i.e., α = a + √ a + 4 b β = a − √ a + 4 b . (2.6)We restrict our objects of interest to real and distinct solutions α, β ∈ R , α (cid:54) = β , implyinga positive discriminant a + 4 b > α = β = a/ α, β ∈ C are not tackled here. The rootsobey the relations α + β = a, αβ = − b, (2.7)(1 − α )(1 − β ) = 1 − a − b, ( d − α )( d − β ) = d − ad − b, (2.8) α = α a + b, β = β a + b, α + β = a + 2 b, (2.9)( α + 1)( β + 1) = a + ( b + 1) . (2.10)Hereby the first couple of relations is identical to Vieta’s theorem on quadratic equations.As elaborated in many introductory textbooks on difference equations (usually by meansof a single worked-through numerical example), the general (or complete) solution of theexponentially nonhomogeneous difference equation (2.1), subject to four constants and twoinitial values, is given by G n = G n ( a, b, c, d ; G , G ) = G ( h ) n + G ( p ) n , (2.11)with integer indices n ∈ Z , a homogeneous solution (for c =0) G ( h ) n = A α n + B β n (2.12)and a particular solution (for c (cid:54) =0, found by inserting a trial term G ( p ) n = pd n into the recur-rence relation) G ( p ) n = c d d − a d − b d n =: p d n ( d − a d − b (cid:54) = 0) . (2.13)We note that our inquiry is restricted to parameter values satisfying the condition d − a d − b =( d − α )( d − β ) (cid:54) = 0 (restriction 2). Degenerate cases with d = α or d = β can be dealt with ina follow-up study.The constants A and B are formally specified with respect to the initial values G and G :the corresponding general solutions G n = A α n + B β n + p d n with n =0 or n =1 constitute alinear system of equations for A and B , namely G = A + B + p and G = Aα + Bβ + p d ,yielding A = G − p d − ( G − p ) βα − β = H − H βα − β (2.14) B = − G − p d − ( G − p ) αα − β = − H − H αα − β , (2.15)with α − β = √ a + 4 b > H n ≡ G n − p d n = G n ( a, b, , H , H ) (2.16)3re but (real valued) Horadam numbers, obeying the recursion relation H n = a H n − + b H n − , (2.17)with initial values H , H and with the explicit solution given by equation (2.12), togetherwith constants A and B as above. Actually, in equation (2.12), G ( h ) n = H n . For a matrixrepresentation of G n in terms of H n , and for a related decomposition, we refer to the Appendix.To summarize, equations (2.11)-(2.15) constitute a fully specified explicit solution G n = A α n + B β n + p d n = H n + p d n (2.18)for the implicit relation (2.1). For the special value d =1 one relies upon the solution of anonhomogenous second-order recurrence relation with constant coefficients. We finally notethat on defining γ := max {| α | , | β | , | d |} for the dominant characteristic root or base (withdistinct α , β , and d ), the dominant solution in the very large- n -limit behaves as n (cid:29) G n (cid:40) ∝ γ n ( γ ≥ ≈ γ < . (2.19)2.2. A selected identity.
Selected sums and identities involving G n are given by Phadte &Pethe (2013) in [33] and by Phadte & Valaulikar (2016) in [34]. A couple of their results withrespect to summations are reformulated in our notation in the Appendix. Here we extend theirwork with an additional nonlinear identity and in Section 2.3 with a novel summation formulathat will be applied in Section 3. By insertion of the solution formula for G n (equ. 2.18) it isa straightforward algebraic exercise to obtain the following product difference identity G n + u G n + v − G n + u + v G n = ( − n +1 b n AB ( α u − β u )( α v − β v )+ p d n + u + v (cid:26) G n + u d − u + G n + v d − v − G n + u + v d − u − v − G n (cid:27) . (2.20)Inserting the conjoined fundamental Horadam numbers h n = G n ( a, b, ,
0; 0 ,
1) = ( α n − β n ) / ( α − β ) (i.e., p = 0 and with initial values 0 and 1, corresponding to the Lucas sequenceof the first kind), the first term on the right-hand side may be reexpressed as( − n +1 b n AB ( α u − β u )( α v − β v ) = ( − b ) n ( H − H α )( H − H β ) h u h v . (2.21)For the special case of Horadam numbers H n = G n ( a, b, , H , H ) (i.e., with p = 0), identity(2.20) was already presented together with relation (2.21) in an equivalent form by Horadam(1987) [19], [20]. Alternatively, adopting the notation in terms of transformed Horadam num-bers (equation 2.16), relation (2.20) may succinctly be written as H n + u H n + v − H n + u + v H n = ( − b ) n (cid:26) H u H v − H u + v H (cid:27) . (2.22)This elegant product difference identity of order two for generalized Fibonacci numbers seemsto have been formulated first by Shannon (1988) in [36] (Lemma 2.3) for Horadam num-bers and henceforth is termed Shannon identity . It obviously constitutes a generalisation ofthe well-known Tagiuri identity F n + u F n + v − F n + u + v F n = ( − n F u F v for Fibonacci num-bers F n = G n (1 , , ,
0; 0 ,
1) ([37],[10],[38]). For properly chosen indices u and v , relation(2.22) subsumes generalized versions of, for example, the Catalan identity ( u = − v ) and thed’Ocagne identity ( u = m − n , v = 1) (e.g., [26]). Here we even allow for real valued andtransformed numbers H n = G n − pd n . In a reversed presentation of a proof, relation (2.20) forthe generalized Fibonacci numbers follows from Shannon’s identity for Horadam numbers by4ENERALIZED FIBONACCI SPIRALsimply substituting the transformation (2.16). Actually, all known identities for (pure) Ho-radam numbers H n must hold for the substitution G n − pd n as well, thus indirectly providinga variety of available identities and summations for the generalized Fibonacci numbers G n .For example, the higher-order identity for G n − b G n − or the weighted sum (cid:80) n − k =1 ( − b ) k G n could be found by means of the corresponding identity and sum formulae for Horadam num-bers, given in [18] as equations (4.25) and (4.26), respectively. An actual application of thissubstitution method is provided in Section 6.3.2.2.3. Sums of alternating even- or odd-indexed terms.
Closed-form expressions forthe sums (cid:80) nk =0 G k and (cid:80) nk =0 ( − k G k are given in the Appendix for later use. The sums (cid:80) nk =0 G k and (cid:80) nk =0 G k +1 could easily be deduced by means of the substitution methodsuggested above upon using the corresponding results for Horadam numbers ([18], equations3.12 and 3.13). Additionally, in order to prepare some visualization of the sequence { G n } bymeans of rectangular and curvy spirals, another couple of sums of differences is calculatedhere.2.3.1. Formalism.
We are concerned with the sum of the first alternating even-indexed termsup to term G n ( n even), (cid:80) n/ k =0 ( − k G k , and with the sum of the first alternating odd-indexed terms up to term G n ( n odd), (cid:80) ( n − / k =0 ( − k G k +1 . Written compactly, we seekΓ n ≡ ( n − ν ) / (cid:88) k =0 ( − k G k + ν , ν = n mod ∈ { , } . (2.23)The mutual summation function Γ n —with index variable n and with an index-dependentand thus constrained binary parameter ν that rules the case distinction between even-indexedterms ( ν =0) and odd-indexed terms ( ν =1)— can be expressed in closed form asΓ n = 1 a + ( b + 1) (cid:26) ( − ( n − ν ) / (cid:16) G n +2 + b G n (cid:17) + G ν + b G ν − − c d + ad − bd + 1 (cid:18) ( − ( n − ν ) / d n +2 + d ν (cid:19) (cid:27) , ν = n mod , (2.24)where G − and G − are given by equations (2.3)f or simply are G n evaluated for n = − n = −
2, respectively. A couple of proofs is relegated to the Appendix.Consequently, the simple difference relationΓ n − Γ n − = ( − ( n − ν ) / G n (2.25)holds, as can easily be checked by means of either definition (2.23) or expression (2.24). Inthe very large- n -limit, one finds due to relation (2.19) n (cid:29) n ∝ ( − ( n − ν ) / γ n +2 ( γ ≥ ≈ a +( b +1) (cid:26) G ν + b G ν − − c d + ad − bd +1 d ν (cid:27) ( γ < . (2.26)We note that the ( γ ≥ γ < ν -dependent) value. 5.3.2. Examples.
Some special cases for Γ n are given now by means of providing algebraic andnumerical examples, hereby partly relying upon integer values for G n . Firstly, the sum of the alternating even- or odd-indexed Fibonacci numbers F n = G n (1 , , ,
0; 0 ,
1) is calculated withΓ
Fibo n = 15 (cid:26) ( − ( n − ν ) / (cid:0) F n +2 + F n (cid:1) + F ν + F ν − (cid:27) , (2.27)where F − = F = 1 and F − = − F = − F − n = ( − n +1 F n .For example, taking n = 9 (hence, ν = 1) one has (cid:80) k =0 ( − k F k +1 = F − F + F − F + F =1 − −
13 + 34 = 25 that is equal to Γ
Fibo = (1 / F + F + F + F − ) = (1 / alternating even- or odd-indexed Horadam numbers H n = G n ( a, b, , H , H ) (including the Lucas, the Pell, or some other famous sequences) can becalculated by means ofΓ Hora n = 1 a + ( b + 1) (cid:26) ( − ( n − ν ) / (cid:16) H n +2 + b H n (cid:17) + H ν + b H ν − (cid:27) , (2.28)with H − = ( H − aH ) /b and H − = (( a + b ) H − aH ) /b according to the negation rules(2.3)f. For example, specifying a =1 and b =1 and renaming the two initial values will reproducethe particular sum formulas given in [40] by Walton & Horadam (1974, equations 4.11f). Wenote that knowing relation (2.28) in advance (as was not the case with respect to the author),deriving the summation formula (2.24) is basically a matter of substitution only. In order toexemplify the substitution method, we correspondingly perform a second proof of (2.24) inthe Appendix.To summarize this section: Having put the two restrictions a +4 b> d − a d − b (cid:54) =0,identity (2.20) and (2.22) and series (2.23) together with summation function (2.24) hold forall integer as well as all real solutions (2.11)-(2.15) of the defining relation (2.1).3. Generalized Fibonacci spirals
Classification of spirals.
Based on the generalized sequence discussed in the previoussection, the following twofold ad-hoc classification is provisionally suggested for the variety ofcorresponding spiral representations:I. Algebraic structure. Refering to frequently heard modes of expression and based on thesecond-order recurrence relation, we may distinguish four types of sequences and correspondingspirals according to a growing degree of generalization:(i) the initial or genuine
Fibonacci numbers G n (1 , , ,
0; 0 ,
1) constitute a correspondingordinary or genuine Fibonacci spiral .(ii) The generalized genuine
Fibonacci numbers G n (1 , , , G , G ) with any real valuedpair of initial values G , G . This results in generalized genuine Fibonacci spirals .(iii) The generalized bivariate Fibonacci numbers G n ( a, b, , G , G ) with real coefficients a , b and real initial values G , G . In the context of integer coefficients and integerinitial values, these are often called Horadam numbers . The numbers G n ( a, b, ,
0; 0 , bivariate Fibonacci spirals or Horadamspirals .(iv) The generalized Fibonacci numbers G n ( a, b, c, d ; G , G ) with an exponential input .For brevity this could be termed ”exponacci numbers”. The corresponding spiralsmay then be called exponentially generalized Fibonacci spirals or exponacci spirals .6ENERALIZED FIBONACCI SPIRALFor short, cases (ii) to (iv) may all be addressed as generalized Fibonacci sequences , irrespectiveof being either integer or real sequences. Correspondingly, there are then three types of generalized Fibonacci spirals .II. Geometric composition. A second class of classification concerns the geometric construc-tion of the spirals and hence their visual appearance. As will be evident, we distinguish(i) shapes between rectangular , angular , and arched spirals, all of which are composed ofelementary graphs added together. While rectangular and angular spirals are drawnby connecting successive points with straight lines, arched spirals are composed ofassembled quarter-circles (relying on cases I.i and I.ii from above) or with quarter-ellipses (based on cases I.iii and I.iv). Another type of curved spiral is a spiral givenin polar form ; it is not based on a discret sequence but on a continuous function andwill be treated in section 4. A further geometric characteristics involves(ii) winding , whereby outwinding means spiraling outwards and inwinding means spiralinginwards. The formal criterion for this distinction is due to the characteristic value γ := max {| α | , | β | , | d |} introduced above, γ ≥ outwinding , γ < inwinding . (3.1)For the first case, some parameter values may allow for some inward spiraling for smallvalues of n (as long as Aα n + Bβ n , with α < β <
1, dominates the values for G n ), followed by outward spiraling for n large enough (as soon as pd n , with d > G n ). We note that if γ = 1, the spiral approaches an attractor of quadraticshape and exhibits cyclic behaviour. The second case ( γ <
1) corresponds to thecombined condition d < < b < − a / α < β < orientation . Typically, negative pa-rameter values a or b may govern rightward orientation.For example, one may have a leftwardly outwinding rectangular spiral (as in the left panel ofFigure 1, black line) or a leftwardly inwinding arched spiral (right panel, red line).3.2. Rectangular spirals.
Corner points.
Rectangular genuine Fibonacci spirals are drawn in a two-dimensionalCartesian coordinate system by means of partitioned spiral arms, whose successive linearsegments meet at right angles and with lengths equal to successive Fibonacci numbers F n = G n (1 , , ,
1; 0 ,
1) (see Figure 1 for the construction scheme and Figure 2 (upper left panel,black line) for a true-to-scale map). The corresponding corner points P n = ( X n ; Y n ) ( n ∈ N )are P = ( F ; 0), P = ( F ; F ), P = ( F − F ; F ), P = ( F − F ; F − F ), and so on. Ingeneral, the coordinates of the corner points are given by sums of alternating even- or odd-indexed Fibonacci numbers F n according to the entries in Table 1 (top). At every corner pointthere is a turn to the left by 90 ◦ , and the distance between any two successive corner pointsis equal to the related Fibonacci number, i.e., P n − P n = F n .Similarly, using generalized Fibonacci numbers G n and adopting relations (2.23) and (2.24)for Γ n , the corner points of a rectangular generalized Fibonacci spiral are located at coordinatesas listed in Table 1 (bottom). In general, the position of the n -th corner point is given by P n = (cid:0) X n ; Y n (cid:1) = (cid:40) (cid:0) Γ n ; Γ n − (cid:1) n even (cid:0) Γ n − ; Γ n (cid:1) n odd . (3.2)7 igure 1. Construction of rectangular and arched generalized Fibonacci spirals (drawn inblack and red, respectively), with inserted denotations for partitioned arm lengths G n (givenby a generalized Fibonacci sequence) and corner points P n (with coordinates determinedaccording to the entries in Tables 1). The rectangular areas used to draw arches by means ofquarter-ellipses are indicated through replenished side lines (red dashed lines); these rectanglesconstitute at the same time a basic tiling pattern. Outwinding spirals are shown in the leftpanel, inwinding spirals in the right panel. We note that all coordinates X n go with ν = 0 and all Y n have ν = 1. This reflects the spiral’sprinciple of construction with the x - and y -coordinates of the corner points being calculatedby differences with even-indexed and odd-indexed values of G n , respectively. By construction,the distance relation P n − P n = G n (3.3)holds. This can formally be checked by inserting equations (2.23) and (3.2) into P n − P n =[( X n − X n − ) + ( Y n − Y n − ) ] / . At every corner point there is a turn to the left by 90 ◦ forpositive values of G n (below) or a turn to the right by 90 ◦ for negative values of G n . As anillustration, we shown in Figure 2 (upper panels) a couple of leftwardly outwinding rectangulargeneralized Fibonacci spirals (black lines). They are based on the genuine Fibonacci numbers F n = G n (1 , , ,
0; 0 ,
1) and on the generalized Fibonacci numbers G n (0 . , . , , .
9; 3 , angular spiral (not shown).3.2.2. Orthogonal asymptotes.
Assuming throughout this subsection positive initial values G and G , the corner points P , P , P , . . . , P k , . . . ( k ∈ N ) constitute the infinite set of alllower-right corner points. In a similar manner, the sequences { P k } , { P k } , and { P k } form the sets of upper-right, upper-left, and lower-left corner points, respectively (see Figure1). In general, one has sets of directional corner points { P j +4 k } , where k ∈ N is the runningnumber and the constant value j ∈ { , , , } fixes the direction. Stated differently, anycorner point P n corresponds to a directional corner point with directional index j = n mod P n and P n +4 (implying an equal index value j and with coordinates as given by equation 3.2) is determined8ENERALIZED FIBONACCI SPIRAL Rectangular Fibonacci spiral P n X n Y n P F P F F P F − F F P F − F F − F P F − F + F F − F P F − F + F F − F + F P F − F + F − F F − F + F P F − F + F − F F − F + F − F Rectangular generalized Fibonacci spiral P n X n Y n P G P G G P G − G G P G − G G − G P G − G + G G − G P G − G + G G − G + G P G − G + G − G G − G + G P G − G + G − G G − G + G − G P n (cid:40) n even n odd (cid:40) Γ n Γ n − (cid:40) Γ n − Γ n Table 1.
Coordinates of the first few corner points P n = ( X n ; Y n ) for rectangularFibonacci spirals and for rectangular generalized Fibonacci spirals. For the generalcase, the involved summation function Γ n is given in closed form by equations (2.23)and (2.24). See Figure 1 for an illustration. Arched Fibonacci spiral: quarter-circlesarc e x e y P F F n ≥ P n − F n F n Arched generalized Fibonacci spiral: quarter-ellipsesarc e x e y P G − G G outwinding: (cid:40) n even n odd P n − (cid:40) G n G n +1 − G n − (cid:40) G n +1 − G n − G n inwinding: (cid:40) n even n odd P n +4 (cid:40) G n +4 − G n +2 G n +3 (cid:40) G n +3 G n +4 − G n +2 Table 2.
Quarter ellipses for arched generalized Fibonacci spirals: the center points and lengths of the semi-axes e x and e y in x - and y -direction, respectively, for the n tharc. The center points of the arcs coincide with particular corner points P i for rect-angular spirals (given in Table 1). For an illustration, see Figure 1. Arched Fibonaccispirals represent a special case with F n = G n (1 , , ,
0; 0 , F n +1 − F n − = F n , their quarter ellipses reduce to quartercircles with fixed radii F n . Figure 2 (upper left panel) provides an illustration. igure 2. Examples of rectangular and arched generalized Fibonacci spirals.
Upper left : Genuine Fibonacci spirals tracing the Fibonacci numbers F n = G n (1 , , ,
0; 0 , n ∈ N ) by means of either perpendicular arms with lengths 0, 1, 1, 2, 3, 5, 8, 13, . . . (blacklines) or successive quarter-circles with corresponding radii (red). The oblique asymptotes forthe directional corner points (blue dashed lines) are orthogonal, with slopes Φ ≈ .
618 (i.e.,the golden ratio) and − / Φ ≈ − . P ∗ = ( − /
5; 2 / Upperright : Outwinding generalized Fibonacci spirals with G n (0 . , . , , .
9; 3 , P ∗ ≈ (1 . . P P and P P ). Lower left : Inwinding generalized Fibonacci spirals with G n (0 . , . , , .
8; 19 , P ∗ ≈ (10 .
63; 8 . Lower right : Outwinding generalized Fibonacci spirals for G n (0 . , . , , .
1; 3 , d with respect to the upper right figure): becausenow α > d >
1, the points of intersection for the asymptotes (here represented bythe dashed lines P P , P P , P P , and P P ) drift away from P (cid:63) (central circle).The other four circles in the vicinity give the positions of approximately calculated points ofintersection (see text for details). Y n +4 − Y n X n +4 − X n = (cid:40) Γ n +3 − Γ n − Γ n +4 − Γ n n even Γ n +4 − Γ n Γ n +3 − Γ n − n odd . (3.4)In the very large n -limit this becomes n (cid:29) Y n +4 − Y n X n +4 − X n → (cid:40) − /γ n even γ n odd . (3.5) Proof.
Assuming Γ n +4 (cid:29) Γ n , Γ n +3 (cid:29) Γ n − (for n (cid:29)
1) and adopting relation (2.26) one has Y n +4 − Y n X n +4 − X n = Γ n +3 − Γ n − Γ n +4 − Γ n ≈ Γ n +3 Γ n +4 ≈ ( − ( n +3 − / γ n +5 ( − ( n +4 − / γ n +6 = − γ n even Γ n +4 − Γ n Γ n +3 − Γ n − ≈ Γ n +4 Γ n +3 ≈ ( − ( n +4 − / γ n +6 ( − ( n +3 − / γ n +5 = γ n odd . Assuming instead Γ n +4 (cid:28) Γ n , Γ n +3 (cid:28) Γ n − (for n (cid:29)
1) one similarly gets Y n +4 − Y n X n +4 − X n = Γ n +3 − Γ n − Γ n +4 − Γ n ≈ Γ n − Γ n ≈ ( − ( n − − / γ n +1 ( − ( n − / γ n +2 = − γ n even Γ n +4 − Γ n Γ n +3 − Γ n − ≈ Γ n Γ n − ≈ ( − ( n − / γ n +2 ( − ( n − − / γ n +1 = γ n odd . (cid:3) This asymptotic behaviour immediately implies the following proposition.
Proposition 3.1.
For rectangular generalized Fibonacci spirals, the oblique asymptotes for thedirectional corner points with either even- or odd-numbered indices lie mutually orthogonal.Proof.
In general, the (complanary) graphs of two linear functions with slopes s and s areperpendicular with respect to each other, if the condition s s = − − /γ and γ . (cid:3) As special case, the rectangular genuine Fibonacci spiral goes with asymptotic slopes throughdirectional corner points that are equal to the Golden ratio and its negative inverse, i.e., γ = lim n →∞ F n +4 /F n +3 = lim n →∞ F n +1 /F n = Φ = 1 . . . . ( n odd) and − /γ = − Φ − = − . . . . ( n even). For its exhibition, see Figure 2 (upper left panel). The two asymptotesdrawn lead through the directional corner points with n =48 and 52 and with n =49 and 53(representing some large- n -limit) and intersect at the point P (cid:63) = ( − /
5; 2 / c = 0) the corresponding asymp-totic slopes are γ = α and − /γ = − /α (equivalent to an eigenvalue of the Horadam matrix(6.1) refered to in the Appendix).Proposition 3.1 holds for both types of windings, i.e., for inwinding and outwinding spirals,however, with some particularities, as emphasized by the following proposition. Proposition 3.2.
For outwinding rectangular spirals with either γ = α > > d > or γ = d > > α > and for inwinding spirals ( γ < ) the two perpendicular asymptotes fordirectional corner points cross at the point of intersection P (cid:63) = ( X (cid:63) ; Y (cid:63) ) given by P ∗ = 1 a + ( b + 1) (cid:16) G + b G − − c d + ad − bd + 1 ; G + b G − − c d d + ad − bd + 1 (cid:17) ; (3.6) for inwinding spirals this coincides with the point of convergence lim n →∞ P n . roof. (Sketch) The claimed coordinates for the point of convergence of inwinding spiranglesdirectly follow by inserting relation (2.26) (case γ <
1) into equation (3.2). Due to Proposition3.1 this must be equal to the point of intersection. For outwinding spirangles with γ > P n = (Γ n ; Γ n − ) ( n even, and for the moment with directional index j =0) and P n +1 = (Γ n +1 ; Γ n ) ( n + 1 odd, j =1) lie in the very large- n -limit on orthogonallines described by linear functions y = − γ − x + (Γ n − + γ − Γ n ) and y = γ x + (Γ n +1 − γ Γ n ), respectively. Hereby the slopes are chosen according to Proposition 3.1. Equating thefunctions yields x ≡ X (cid:63)n,j =0 and y ≡ Y (cid:63)n,j =0 as given by X (cid:63)n,j =0 = Γ n − γ ( γ + 1) − (Γ n +1 − Γ n − ) = Γ n − γ ( γ + 1) − G n +1 and Y (cid:63)n,j =0 = − γ − X (cid:63)n + (Γ n − + γ − Γ n ) = Γ n − + ( γ +1) − G n +1 . Adopting either γ = α > > d or γ = d > > α and taking G n ≈ Aα n or G n ≈ pd n , respectively (due to relation 2.26), inserting expression 2.24 and using relations(2.6)-(2.8) straightforwardly provides the reclaimed point P (cid:63) . Similarly, if one starts theinquiry with directional indices j =1,2,3 (instead of j =0 as above), one gets approximatepoints of intersection ( X (cid:63)n,j =1 , Y (cid:63)n,j =1 ) = (Γ n − ( γ + 1) − G n +2 ; Γ n +1 − γ ( γ + 1) − G n +2 ),( X (cid:63)n,j =2 , Y (cid:63)n,j =2 ) = (Γ n +2 + γ ( γ + 1) − G n +3 ; Γ n +1 − ( γ + 1) − G n +3 ), and ( X (cid:63)n,j =3 , Y (cid:63)n,j =3 ) =(Γ n +2 + ( γ + 1) − G n +4 ; Γ n +3 + γ ( γ + 1) − G n +4 ), respectively. Adopting the procedure asabove for j = 0, the conclusion arrived before concerning P (cid:63) remains the same in all cases. (cid:3) As with the rectangular Fibonacci spiral shown in Figure 2 (upper left panel), the asymp-totes for the rectangular generalized Fibonacci spirals shown in the upper right panel (out-winding) and in the lower left panel (inwinding) are approximated by the blue-dashed linesgoing through points P (and P ) and P (and P ). In all examples the point of intersection P (cid:63) is calculatd with equation (3.6) and marked by a blue circle.For differently outwinding rectangular spirals, i.e., those with both α > d >
1, theslopes of lines through two neighbouring directional corner points still obey Proposition 3.1.But in this case the mutual points of intersection P ∗ n,j = ( X (cid:63)n,j ; Y (cid:63)n,j ) —as given approximatelyin the proof above— do not converge, with increasing value of n they instead diverge awayfrom P (cid:63) . A zoom-in illustration is given in the lower right panel of Figure 2. The pointsof intersection (crossing dashed lines) are loacated only in the vicinity of P (cid:63) (single centralcircle). Their drift distances are, however, much smaller than the respective coordinates ofthe directional corner points: if the axes in this lower right panel would be scaled-out to thevalue of, say, P = (17478; − P (cid:63) invisible close. Thereader is invited to check this herself with the help of a graphical tool.These mutually orthogonal lines are closely related to what may be called Holden lines ,because a similar feature of orthogonality was already recognized by Holden (1975) [16] for thesuccessive centers of quadratic tiles related to an outwinding genuine Fibonacci spiral. Hoggatt& Alladi (1976) [15] modified the results of Holden by means of an inwinding rectangularHoradam spiral defined by the sequence G n (1 , , ,
0; 1 , G ) and found the intersection of theasymptotes of the directional corner points to correspond to the point of convergence for theirspiral. Except for a rotation of the coordinate system, their observation seems to encounter aspecial case of our generalized approach.Last but not least, we note that the length L n of a rectangular generalized Fibonacci spiralthat starts at the origin of the coordinate system can be calculated by either one of the sumformulae (6.9) to (6.11). In case of an inwinding spiral with infinitely many segments, thefinite total length becomes L ∞ = 1 a + b − (cid:32) ( a − G − G − cd − d (cid:33) . (3.7)12ENERALIZED FIBONACCI SPIRAL3.3. Arched spirals.
As can be inferred from Figure 1 and Table 2, the starting point of the n -th quarter-circle of an arched Fibonacci spiral has radius F n and the center is located at cornerpoint P n − (with the coordinates according to equation 3.2). Modestly more complicated, the n -th quarter-ellipse of an arched generalized Fibonacci spiral has unequally long semi-axesand depends on the even/odd distinction for n . As with arched genuine Fibonacci spirals,the center is identical with corner point P n − for outwinding spirals. However, for inwindingspirals the center is chosen here to be identical with corner point P n +4 . (Alternatively, anotherinviting choice would have been P n +6 .) In order to draw the arc of the n -th quarter-ellipse or-circle, some N + 1 spiral points P n,i , i = 0 , , , . . . , N , with Cartesian coordinates ( x n,i ; y n,i )on the arc are calculated and interpolated. We use parameter representation to have spiralpoints P out n,i = ( x n,i ; y n,i ) = (cid:40) (cid:0) Γ n − + G n cos φ n,i ; Γ n − + ( G n +1 − G n − ) sin φ n,i (cid:1) n even (cid:0) Γ n − + ( G n +1 − G n − ) cos φ n,i ; Γ n − + G n sin φ n,i (cid:1) n odd , (3.8)for outwinding spirals and P in n,i = ( x n,i ; y n,i ) = (cid:40) (cid:0) Γ n +4 + | G n +4 − G n +2 | cos φ n,i ; Γ n +3 + G n +3 sin φ n,i (cid:1) n even (cid:0) Γ n +3 + G n +3 cos φ n,i ; Γ n +4 + | G n +4 − G n +2 | sin φ n,i (cid:1) n odd , (3.9)for inwinding spirals, in both cases with attributed polar angles φ n,i = (cid:18) n + iN (cid:19) π ∈ (cid:20) n π , ( n + 1) π (cid:21) , i = 0 , , , . . . , N. (3.10)For our figures, we choose N = 60. By definition, the starting points of the first and of thesecond arc are P , = (0; G ) and P , = ( G − G ; 0), respectively. (This may be modified inanother study.) For n ≥ φ n, = n π/ i = 0 in equation 3.9), the starting point ofthe n -th arc is P n, = ( x n, ; y n, ) = (cid:40) (cid:0) Γ n − + G n ( − n/ ; Γ n − (cid:1) = (cid:0) Γ n ; Γ n − (cid:1) n even (cid:0) Γ n − ; Γ n − + G n ( − ( n − / (cid:1) = (cid:0) Γ n − ; Γ n (cid:1) n odd , (3.11)where the rear equalities directly follow from equation (2.25).The assemblage of arcs of subsequent quarter-ellipses composes an arched generalized Fi-bonacci spiral in the plane. Some example spirals, including the popular planar Fibonaccispiral, are shown in Figure 2 (red curves).Some further observations concerning large values of n are as follows: • As already stated above, for large n the value of G n = A α n + B β n + p d n ( α , β , d being distinct) is dominated by γ = max {| α | , | β | , | d |} , i.e., G n ∝ γ n . • The ellipticity of the n th arc (with n even) is defined by (cid:15) out = 1 − ( G n +1 − G n − ) /G n for outwinding spials and (cid:15) in = 1 − G n +3 / | G n +4 − G n +2 | for inwinding spirals. Forlarge n , the ellipticities approach constant values 1 − γ and 1 − /γ , respectively. Asimilar statement can be made for the case with n being odd. • For large n , the spiral points are approximated with increasing accuracy by n large : P n,i ≈ (cid:40) (cid:0) Γ n − + γ n cos φ n,i ; Γ n − + γ n +1 sin φ n,i (cid:1) n even (cid:0) Γ n − + γ n +1 cos φ n,i ; Γ n − + γ n sin φ n,i (cid:1) n odd . (3.12)13 igure 3. Spacial generalized Fibonacci spiral.
Left : Perspective view of the outwindingspiral shown in Figure 2 (upper right panel, reproduced here in red), with the z -coordinategiven by the cumulated inputs according to equation (3.15). Right : Same spiral as shown inthe panel to the left, but seen projected onto the x - z -plane in a frontal view. For n → ∞ theheight of the spiral will converge to the upper limit z ∞ = c/ (1 − d ) = 10 (with c = 1 , d = 0 . For Fibonacci numbers F n = G n (1 , , ,
0; 0 ,
1) or Lucas numbers L n = G n (1 , , ,
0; 2 , γ = Φ ≈ .
618 being equal to the golden ratio, this is known to approxi-mately become a particular geometric spiral (a.k.a. logarithmic or Bernoulli spiral).Such a spiral is investigated in, e.g., [9].3.4.
Spacial representation.
We finally add a third dimension to arched spirals to get in-terpolated points with spacial coordinates P n,i = ( x n,i ; y n,i ; z n,i ), with x n,i , y n,i still calculatedaccording to equations (3.8)-(3.11). For the third coordinate z n,i , i = 0 , , , . . . , N −
1, onemay simply choose some linear increase, i.e., z n,i = n + iN , (3.13)where N is the number of points used to interpolate each quarter-ellipse. Or one may visualizethe exponential input that appears in recursion relation (2.1) by exhibiting either the localcontribution z n,i = c d n + i/N (3.14)or the cumulated inputs z n,i = c d n +1+ i/N − d − , (3.15)where for i = 0 one has in the latter case z n,i = c S n ( d ), S n ( d ) being the partial sum formula forgeometric progression (see equation 6.8). Alternatively, replacing c by p will provide an alteredinformation. A typical spacial spiral is exhibited in Figure 4, seen under distinct perspectives.It is but the outwinding arched spiral already shown in Figure 2 (upper right panel), with the z -coordinate calculated according to the rule given by equation (3.14). Because G n is growingwith increasing index number n , the spiral arms get exponentially broader, while the heightof this spiral approaches an upper limit (see Figure caption).14ENERALIZED FIBONACCI SPIRAL Figure 4.
Illustrating graphs in the Gaussian plane for the complex valued function G ( t | a, b, c, d ; G , G ) with real argument t (according to equation 4.3). The small circles at theintersection with the axis of the real part, i.e., at the zeros of the function Im [ G ( t )], provide thevalues of the sequence { G n ( a, b, c, d ; G , G ) } . This occurs for all integer values of the variable,i.e., for t = n ∈ N . Upper left:
Oscillatory graph for G ( t | . , . , . , . − . , . < b/α < Upper right:
Spiral graph with G ( t | − . , . , . , . − . , . b/α > Lower left:
Examples with integer parameter values, once for G ( t | , , , − , −
2) (thin line, b/α > G ( t | , , ,
0; 0 ,
1) (thick line, 0 < b/α < B ≈ .
417 in equ. 4.3), in the latter case there is damped oscillatory be-haviour. The integer Fibonacci numbers appear at the intersection of the graph with the realpart axis, i.e., F n = G ( t | t = n ). Lower right:
Spiraling graph for G ( t | − , , ,
0; 0 ,
1) (corre-sponding to b/α > alternating
Fibonacci numbers at the zeros of the imaginarypart.
Generalized Fibonacci spirals with real argument
So far, the index n has been an integer number producing a real valued sequence { G n } .Replacing now the index n by some real argument t , we have the complex valued function G ( t ) = G ( t | a, b, c, d ; G , G ) = A α t + B β t + p d t , t ∈ R +0 (4.1)with A and B still given by equations (2.14) and (2.15), respectively. Indeed, applying Euler’sformula, i.e., ( − t = exp( iπ t ) = cos( π t ) + i sin( π t ), and reminding αβ = − b , one mayexpress G ( t ) = A α t + B ( − b/α ) t + pd t (4.2)= (cid:16) A α t + B cos( πt )( b/α ) t + pd t (cid:17) + i (cid:16) B sin( πt )( b/α ) t (cid:17) (4.3)= Re [ G ( t )] + i Im [ G ( t )] . (4.4)This is analogous to the treatment by Horadam & Shannon (1988) [21] for Fibonacci and Lucasnumbers or by Horadam (1988) [22] for Jacobsthal and Pell numbers. For a similar, but morerecent approach, see, e.g., [6] and [32]. Drawing the generalized Fibonacci function G ( t ) in theGaussian plane creates either oscillatory or spiral curves, depending whether the parameter b/α obeys 0 < b/α ≤ b/α >
1, respectively. The former criterium results in the gradualdisapperance of the trigonometic terms, while the latter criterium guarantees for nonvanishingcontributions of both the cosinus term and the sinus term, providing the circularity of thegraph. At the zeros of Im [ G ( t )] the equality G ( t ) = G n holds (with G n as introduced inSection 2), because for all integer values of t one has cos( π t ) = ± π t ) = 0. Hence,as a corollary , G ( t ) = G n ∀ t ∈ N . (4.5)Figure 4 illustrates this kinship and the general behaviour of the graphs for pairwise nearlyequal sets of parameter values. See figure caption for some details. The oscillating ”Fibonaccicurve” originally shown in [21] is depicted, too (lower left panel, thick line with red circles forthe Fibonacci numbers). We finally note that some spacial representation of G ( t ) could bemanaged in a similar manner as in Section 3.4.5. Conclusions
We draw generalized Fibonacci spirals, based on analytic solutions of the recurrence relation G n = a G n − + b G n − + c d n . The underlying generalized Fibonacci numbers G n are equivalentto transformed Horadam numbers H n = G n − pd n . Our inquiry is restricted to positive realinitial values G and G and coefficients a , b , c , and d , that additionally satisfy the conditions a + 4 b > d − a d − b = ( d − α )( d − β ) (cid:54) = 0 (restriction 2). Complexsolutions for G n (for a +4 b <
0) or degenerate cases with d = α or d = β , as well as allowing fornegative parameter values could be dealt with in a follow-up study. The principal coordinatesused to draw the spirals correspond to finite sums of alternating even- or alternating odd-indexed terms G n and are given in closed-form. For this closed-form solution Γ n (equation2.18), two proofs are given (see Appendix), one based on the Moivre-Binet-form of G n anda shorter one based on the transformed Horadam numbers H n = G n − pd n that adopts thesubstitution method suggested in the text. For rectangular spirals composed of straight linesegments, the even-indexed and the odd-indexed directional corner points asymptotically lieon mutually orthogonal oblique lines. We calculate the points of intersection and show themin the case of inwinding spirals to coincide with the calculable point of convergence. In thecase of outwinding spirals, an n -dependent quadruple of points of intersection may form. We16ENERALIZED FIBONACCI SPIRALillustrate the situation and provide approximate coordinate calculations. For arched spirals,interpolation between principal coordinates is performed by means of arcs of quarter-ellipses. Asimple three-dimensional representation that visualizes the exponential input c d n is exhibited,too. Other choices can be thought of, and extensions up to spiraling surfaces in 3D space seemattractive. The continuation of the discrete sequence { G n } to the complex-valued function G ( t ) with real argument t ∈ R exhibits spiral graphs and oscillating curves in the Gaussianplane, thereby subsuming the values G n for t ∈ N as the zeros.Besides, we retrieve within our framework the Shannon identity (i.e., a generalization of theTagiuri product difference Fibonacci identity) and suggest the substitution method in orderto find a variety of other identities and summations related to G n . This may pose problemssuitable for the problem’s section of The Fibonacci Quarterly [42]. The Shannon identity forHoradam numbers in particular could provide helpful in looking for more and higher-orderproduct identities for H n and hence for G n . This would be in the spirit of former inquiries likethose of Melham [28], [29] and many others (e.g., [11]) in the case of Fibonacci and Horadamnumbers. For that purpose, it could be advantageous to rely on the matrix representation of G n in terms of transformed Horadam numbers (as provided in the Appendix), and to adoptmethods as outlined in, e.g., [39], [24], [4]. Finally, it is to hope that the generalized Fibonaccispirals prove good for some artistic, technical, natural, or even extra-terrestrial applications.6. Appendix
Matrix representation.
Because the numbers G n are but transformed Horadam num-bers H n (equ. 2.16), the system matrix for recurrent (real valued) Horadam numbers H = (cid:18) a b (cid:19) (6.1)—called R-matrix in [39]— does represent recurrence relation (2.1), too, according to (cid:18) H n H n − (cid:19) = (cid:18) G n − pd n G n − − pd n − (cid:19) = H (cid:18) G n − − pd n − G n − − pd n − (cid:19) = H n − (cid:18) G − pdG − p (cid:19) = H n − (cid:18) H H (cid:19) . (6.2)Its eigenvalues are α and β as given by equation (2.6) and its eigenvectors are (cid:18) α (cid:19) and (cid:18) β (cid:19) . Defining matrices D = (cid:18) β α (cid:19) and T = (cid:18) β α (cid:19) , with D n = (cid:18) β n α n (cid:19) and T − = α − β (cid:18) − α − β (cid:19) , allows for diagonalization according to H = T DT − . Hence, H n = T DT − T DT − . . . T DT − = T D n T − or H n = 1 α − β (cid:18) α n +1 − β n +1 − βα n +1 + αβ n +1 α n − β n − βα n + αβ n (cid:19) (6.3)= 1 α − β (cid:18) α n +1 − β n +1 b (cid:0) α n − β n (cid:1) α n − β n b (cid:0) α n − − β n − (cid:1)(cid:19) (6.4)= (cid:18) h n +1 b h n h n b h n − (cid:19) , (6.5)where h n = G n ( a, b, ,
0; 0 ,
1) = ( α n − β n ) / ( α − β ) are the conjoined fundamental Horadamnumbers (i..e., with initial values 0 and 1). Equation (6.2) can easily be recast by meansof equation (6.3). For the Fibonacci numbers F n = G n (1 , , ,
0; 0 , F n = (cid:18) F n +1 F n F n F n − (cid:19) . (6.6)Matrix representations are frequently used in the literature to produce summation identitiesand to establish other properties in relation to recurrent sequences (e.g., recently, [3], [24],[25]). For example, the first component in equation (6.2) implies the decomposition G n = h n G + bh n − G + p (cid:0) d n − dh n − bh n − (cid:1) . (6.7)We do not pursuit the matrix method any further here.6.2. Some partial sums related to { G n } . The main ingredients for the derivations of thesummation formulae given below and in the next subsection are repeatedly the recurrencerelation (2.1), the Moivre-Binet-type solution formula (2.11)ff, and the partial sum formulafor geometric progression with a factor d ∈ R , i.e., S n ( d ) ≡ n (cid:88) k =0 d k = (cid:40) d n +1 − d − ( d (cid:54) = 1) n + 1 ( d = 1) (6.8)Proceeding either as exemplified in Horadam (1965) [18], section 3, by repeated use of therecurrence relation or by insertion of the Moivre-Binet-type solution for G n one straightfor-wardly arrives at n (cid:88) k =0 G k = 1 a + b − (cid:32) ( a − G − G + b G n + G n +1 − cd S n − ( d ) (cid:33) (6.9)= 1 a + b − (cid:32) ( a − G − G n +1 ) − G + G n +2 − cd S n ( d ) (cid:33) (6.10)= ( a − H − H + b H n + H n +1 a + b − pS n ( d ) . (6.11)Herein H n = G n − pd n (equ. 2.16). The three alternative formulations are given for ease ofcomparability with results of different approaches. For example, replacing n → n − A = cd )[34]. For p = 0 one basically recovers the formula originally presented in [18]. Therefore,adopting the substitution method suggested in Section 2.2 and directly starting with identity(6.11) (with p = 0) readily provides some sum formula for G n − pd n and hence for G n .Similarly, the corresponding result for cumulated differences becomes n (cid:88) k =0 ( − k G k = 1 a − b + 1 (cid:18) ( a + 1) G − G + ( − n (cid:16) G n +1 − b G n (cid:17) + cd S n − ( − d ) (cid:19) (6.12)= 1 a − b + 1 (cid:18) ( a + 1) (cid:16) G + ( − n G n +1 (cid:17) − G + ( − n +1 G n +2 + cd S n ( − d ) (cid:19) (6.13)= ( a + 1) H − H + ( − n (cid:16) H n +1 − b H n (cid:17) a − b + 1 + pS n ( − d ) . (6.14)18ENERALIZED FIBONACCI SPIRALFor some more partial sums of linear or quadratic terms regarding sequence G n , see [34]. Thedistinct summation relation given by our equations (2.23)-(2.24) is, however, original work byours and proven below.6.3. Proof for Γ n . In Section 6.3.1, we prove the sum formula (2.23)f for alternating even-indexed or alternating odd-indexed generalized Fibonacci numbers by means of the Moivre-Binet-form solution of G n . In Section 6.3.2, in a second proof of equation (2.23)f, we presumeavailability of the sum formula for the special case of Horadam numbers (equation 2.28) andproceed by means of the substitution method suggested in Section 2.6.3.1. Relying on the Moivre-Binet-form solution.
For a proof of equation (2.23)f we distin-guish in advance the summations both according to n being even or odd and with respect tothe further constraint n mod n even) or n mod n odd). This latterdistinction seems necessary due to the changing occurence of equal or unequal numbers ofsummands with positive or with negative signs (cf. Table 1).The formula for the case n even and n mod n = 4 , , , . . . ) derives in detailas follows (with notations · · · and ( .. ) abbreviating some similar treatments or expressions,respectively): n/ (cid:88) k =0 ( − k G k = n (cid:88) k =0 G k − n − (cid:88) k =0 G k +2 (6.15)= A n (cid:88) k =0 ( α ) k + B n (cid:88) k =0 ( β ) k + p n (cid:88) k =0 ( d ) k − Aα n − (cid:88) k =0 ( α ) k − Bβ n − (cid:88) k =0 ( β ) k − pd n − (cid:88) k =0 ( d ) k = A α n +4 − α − B · · · + p · · · − Aα α n − α − − B · · · − p . . . = A α n +2 ( α −
1) + ( α − α − α + 1) + B · · · + p . . . = A α n +2 + 1 α + 1 + B β n +2 + 1 β + 1 + p d n +2 + 1 d + 1 (6.16)= A ( α n +2 + 1)(1 + β )( d + 1)( α + 1)( β + 1)( d + 1) + B · · · + p . . . = A (cid:18) α n +2 + ( αβ ) α n + 1 + β α α (cid:19) ( d + 1)( .. )( .. )( .. ) + B (cid:18) β n +2 + ( αβ ) β n + 1 + α β β (cid:19) ( d + 1)( .. )( .. )( .. )+ p (cid:18) d n +2 + ( αβ ) d n + 1 + α β d (cid:19) ( d + 1)( .. )( .. )( .. ) − p (cid:18) d n +2 + ( αβ ) d n + 1 + α β d (cid:19) ( d + 1)( .. )( .. )( .. ) 19 p (cid:18) d n +2 + 1 (cid:19) (1 + α )(1 + β ) ( .. )( .. )( .. ) // equ . (2.7) , (2.8) , (2.9)= (cid:18) G n +2 + b G n + G + b G − (cid:19) ( d + 1)( .. )( .. )( .. ) − p ( d n + d − ) (cid:18) d − ( α + β ) d + ( αβ ) (cid:19) ( .. )( .. )( .. )= G n +2 + b G n + G + b G − a + ( b + 1) − p (cid:18) d − ( a + 2 b ) d + b (cid:19) ( d n + d − )( a + ( b + 1) )( d + 1) // equ . (2.13)= 1 a + ( b + 1) (cid:26) G n +2 + b G n + G + b G − − c d + ad − bd + 1 (cid:16) d n +2 + 1 (cid:17) (cid:27) . (6.17)With the last step one observes the polynom division ( d − ( a + 2 b ) d + b ) / ( d − ad − b ) = d + ad − b .Omitting the details, in a very similar way as above one obtains for the case n even and n mod n = 6 , , , . . . ): n/ (cid:88) k =0 ( − k G k = n − (cid:88) k =0 G k − n − (cid:88) k =0 G k +2 (6.18)= − A α n +2 − α − − B β n +2 − β + 1 − p d n +2 − d + 1 (6.19)= 1 a + ( b + 1) (cid:26) − (cid:16) G n +2 + b G n (cid:17) + G + b G − − c d + ad − bd + 1 (cid:16) − d n +2 + 1 (cid:17)(cid:27) . (6.20)For the case n odd and n mod n = 1 , , , . . . ) one has: ( n − / (cid:88) k =0 ( − k G k +1 = n − (cid:88) k =0 G k +1 − n − (cid:88) k =0 G k +3 (6.21)= A α n +2 + αα + 1 + B β n +2 + ββ + β + p d n +2 + dd + 1 (6.22)= 1 a + ( b + 1) (cid:26) G n +2 + b G n + G + b G − − c d + ad − bd + 1 (cid:16) d n +2 + d (cid:17)(cid:27) . (6.23)Finally, the case n odd and n mod n = 7 , , , . . . ) gives ( n − / (cid:88) k =0 ( − k G k +1 = n − (cid:88) k =0 G k +1 − n − (cid:88) k =0 G k +3 (6.24)= − A α n +2 − αα + 1 − B β n +2 − ββ + 1 − p d n +2 − dd + 1 (6.25)= 1 a + ( b + 1) (cid:26) − (cid:16) G n +2 + b G n (cid:17) + G + b G − − c d + ad − bd + 1 (cid:16) − d n +2 + d (cid:17)(cid:27) . (6.26)Drawing a comparison of the results (6.17), (6.20), (6.23), and (6.26) allows to unify for all n according to equations (2.23) and (2.24). (cid:3) Applying the substitution method.
To start with, only even-indexed terms are consid-ered. Inserting the transformation (2.16) into the expression for the series and into equation(2.28) gives Γ
Hora n = n (cid:88) k =0 ( − k H k = n (cid:88) k =0 ( − k G k − p n (cid:88) k =0 ( − k d k = Γ n − p n (cid:88) k =0 ( − d ) k = Γ n − p ( − n/ d n +2 + 1 d + 1 (6.27)and ( a + ( b + 1) Γ Hora n = ( − n/ (cid:16) H n +2 + b H n (cid:17) + H + b H − = ( − n/ (cid:16) G n +2 − pd n +2 + b G n − pb d n (cid:17) + G − p + b G − − pb d − = ( − n/ (cid:16) G n +2 + b G n (cid:17) + G + G − − p d + b d (cid:16) ( − n/ d n +2 + 1 (cid:17) . (6.28)Solving equation (6.27) for Γ n , inserting (6.28), replacing p according to (2.13), and makinguse of the relation ( d + ad − b )( d − ad − b ) = d − ( a + 2 b ) d + b (as in the proof above),Γ n as given in equation (2.24) is easily recovered. To end with, a very similar derivation forthe summation of odd-indexed terms can be accomplished, and the proof is complete. References [1] M. Bicknell-Johnson and G. E. Bergum,
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