Closed Form Equations for Triangular Numbers Multiple of Other Triangular Numbers
aa r X i v : . [ m a t h . G M ] F e b CLOSED FORM EQUATIONS FOR TRIANGULAR NUMBERSMULTIPLE OF OTHER TRIANGULAR NUMBERS
VLADIMIR PLETSER
Abstract.
Triangular numbers that are multiple of other triangular num-bers are investigated. It is known that for any positive non-square integermultiplier, there is an infinity of multiples of triangular numbers which aretriangular numbers. If the multiplier is a squared integer, there is either oneor no solution, depending on the multiplier value. Instead of recurrent rela-tions, we develop in this paper closed form equations to calculate directly thevalues of triangular numbers and their indices without the need of knowingthe previous solutions. We develop the theoretical equations for four cases ofranks from 1 to 4 and we give several examples for non-square multipliers 2,3, 5 and 8.
Keywords : Triangular Numbers, Multiple of Triangular Numbers, Closed FormEquations 1.
Introduction
Triangular numbers are defined as T t = t ( t +1)2 and enjoy many properties, relationsand formulas (see e.g. [1, 2]). Triangular numbers T ξ that are multiples of othertriangular number T t (1) T ξ = kT t have been investigated in the past in some specific cases ([3, 4, 5, 6, 7, 8]). Recently,Pletser showed ([9]) that, first, for square integer values of k , there are either no oronly one solution of (1), depending on the value of k ; and second, for non-squareinteger values of k , there are infinitely many solutions of (1) and recurrent relationscan be found for the four variables t, ξ, T t and T ξ t n = 2 ( κ + 1) t n − r − t n − r + κ (2) ξ n = 2 ( κ + 1) ξ n − r − ξ n − r + κ (3) T t n = (cid:16) κ + 1) − (cid:17) T t n − r − T t n − r + ( T κ − γ ) (4) T ξ n = (cid:16) κ + 1) − (cid:17) T ξ n − r − T ξ n − r + k ( T κ − γ ) (5)These recurrent relations involve three parameters specific to each case of the mul-tiplier k , i.e., a rank r , defined as the number of successive values of t solutionsof (1) with slowly decreasing ratios of the r th to the ( r − th values of t , t r − /t r .The two other parameters are κ and γ respectively the sum κ = t r − + t r and theproduct γ = t r − t r of the first two sequential values of t n for n = r − and r .Several relations exist between these parameters and are investigated in [9]. Notethat only cases with k > are of interest as k = 0 and k = 1 yield obvious solutions respectively ξ = 0 and ξ = t , both ∀ t . Although triangular numbers T t are usuallydefined for t ∈ Z + , triangular numbers can be extended to negative indices t < as T − t = T t − .In this paper, instead of solutions with recurrent relations, we investigate closedform solutions of (1) for the case of non-square integer values of k .2. Closed Form Equations
General equations.
One wishes to calculate directly the n th value of t n , ξ n , T t n and T ξ n without necessarily knowing smaller values needed for recurrent solutions.Closed form equations are easy to calculate from the recursive relations (2) to (5),which are all linear and non-homogeneous. Let us consider first (2). The associatedclosed form equation is the sum of the homogeneous solution t h n and a particularsolution t p (for more details, see e.g. [10, 11, 12]).The homogeneous characteristic equation associated to (2) reads successively x r − κ + 1) x r + 1 = 0 (6) ( x r − α r ) ( x r − β r ) = 0 (7)with α = (cid:16) ( κ + 1) + p κ ( κ + 2) (cid:17) r (8) β = (cid:16) ( κ + 1) − p κ ( κ + 2) (cid:17) r (9)with the obvious relation ( αβ ) r = 1 . This equation (7) has r distinct characteristicroots, including real and complex roots and, for odd r , r = 2 ρ + 1 , one has x r − α r = ( x − α ) ρ X j =0 α j x ρ − j = ( x − α ) ρ Y j =1 (cid:20) x − α (cid:18) cos (cid:18) jπr (cid:19) ± i sin (cid:18) jπr (cid:19)(cid:19)(cid:21) (10)and for even r , r = 2 ρ , x r − α r = ( x − α ) ( x + α ) ρ − X j =0 α j x ρ − j − = ( x − α ) ( x + α ) ρ − Y j =1 (cid:20) x − α (cid:18) cos (cid:18) jπρ (cid:19) ± i sin (cid:18) jπρ (cid:19)(cid:19)(cid:21) (11) LOSED FORM EQUATIONS FOR TRIANGULAR NUMBERS MULTIPLE OF OTHER TRIANGULAR NUMBERS3 and similar relations for β replacing α . Recall that complex characteristic rootsintroduce the element of periodicity. The homogeneous solution is therefore respec-tively (12) for odd r , r = 2 ρ + 1 and (13) for even r , r = 2 ρ , t h n = A + ρ X j =1 (cid:20) A ′ j cos (cid:18) njπr (cid:19) + A ′′ j sin (cid:18) njπr (cid:19)(cid:21) α n (12) + B + ρ X j =1 (cid:20) B ′ j cos (cid:18) njπr (cid:19) + B ′′ j sin (cid:18) njπr (cid:19)(cid:21) β n t h n = A + ( − n A ′ + ρ − X j =1 (cid:20) A ′′ j cos (cid:18) njπρ (cid:19) + A ′′′ j sin (cid:18) njπρ (cid:19)(cid:21) α n (13) + B + ( − n B ′ + ρ X j =1 (cid:20) B ′′ j cos (cid:18) njπρ (cid:19) + B ′′′ j sin (cid:18) njπρ (cid:19)(cid:21) β n with the r constants A, A ′ , A ′′ j , A ′′′ j , B, B ′ , B ′′ j and B ′′′ j to be determined by r boundary conditions, namely the first r values of t n , for n = 0 to n = 2 r − . Inpractice, one can simplify this method by using the fact that t j = t − ( j +1) , i.e. for n = − r to n = r − , necessitating to only know the first ( r − values of t n .The particular solution t p can be found by posing it equal to a constant t p = τ .This particular solution must be a solution of the initial equation (2). Thereforereplacing t n , t n − r and t n − r by t p = τ in (2) yields directly τ = − / . The completesolution is then t n = t h n − / .2.2. Closed form equations for r = 1 ( ρ = 0 ). The homogeneous solution (12)reduces to(14) t h n = Aα n + Bβ n yielding the complete solution(15) t n = Aα n + Bβ n − / with the boundary conditions t = t − = 0 , yielding A + B = 1 / and Aα − + Bβ − = 1 / , giving A = α (1 − β ) / α − β ) and B = β ( α − / α − β ) .2.2.1. Closed forms for k = 2 . The smallest case of k having a rank equal to unityis k = 2 , for which t r − = 0 , t r = 2 (see OEIS [13], A053141) and κ = 2 , yieldingfrom (8) and (9), α = (cid:0) √ (cid:1) = (cid:0) √ (cid:1) , β = (cid:0) − √ (cid:1) = (cid:0) − √ (cid:1) , andfurther, A = (cid:0) √ (cid:1) / , B = (cid:0) − √ (cid:1) / , and the closed form equation(16) t n = 18 (cid:18)(cid:16) √ (cid:17) (cid:16) √ (cid:17) n + (cid:16) − √ (cid:17) (cid:16) − √ (cid:17) n (cid:19) − The closed form equations for the three other variables for this case of k = 2 arefound similarly:- for ξ n (3) (see OEIS [13], A001652)(17) ξ n = 14 (cid:18)(cid:16) √ (cid:17) n +1 + (cid:16) − √ (cid:17) n +1 (cid:19) − LOSED FORM EQUATIONS FOR TRIANGULAR NUMBERS MULTIPLE OF OTHER TRIANGULAR NUMBERS4 - for T t n (4) (see OEIS [13], A075528), with (cid:0) ± √ (cid:1) = (cid:0) ± √ (cid:1) ,(18) T t n = 164 (cid:18)(cid:16) √ (cid:17) n +2 + (cid:16) − √ (cid:17) n +2 (cid:19) − - for T ξ n (5) (see OEIS [13], A029549)(19) T ξ n = 132 (cid:18)(cid:16) √ (cid:17) n +2 + (cid:16) − √ (cid:17) n +2 (cid:19) − Closed forms for k = 3 . For k = 3 , r = 1 and t r − = 0 , t r = 1 (see OEIS[13], A061278) and κ = 1 , yielding α = (cid:0) √ (cid:1) , β = (cid:0) − √ (cid:1) from (8) and (9),and A = (cid:0) √ (cid:1) / , B = (cid:0) − √ (cid:1) / , and the closed form(20) t n = 112 (cid:16)(cid:16) √ (cid:17) (cid:16) √ (cid:17) n + (cid:16) − √ (cid:17) (cid:16) − √ (cid:17) n (cid:17) − Similarly, the closed forms for the three other variables are:- for ξ n (3) (see OEIS [13], A001571):(21) ξ n = 14 (cid:16)(cid:16) √ (cid:17) (cid:16) √ (cid:17) n + (cid:16) − √ (cid:17) (cid:16) − √ (cid:17) n (cid:17) − - for T t n (4) (see OEIS [13], A076139), with (cid:0) ± √ (cid:1) = (cid:0) ± √ (cid:1) :(22) T t n = 148 (cid:18)(cid:16) √ (cid:17) n +1 + (cid:16) − √ (cid:17) n +1 (cid:19) − - for T ξ n (5) (see OEIS [13], A076140):(23) T ξ n = 116 (cid:18)(cid:16) √ (cid:17) n +1 + (cid:16) − √ (cid:17) n +1 (cid:19) − Closed form equations for r = 2 ( ρ = 1 ). The homogeneous solution (13)reduces to(24) t h n = ( A + ( − n A ′ ) α n + ( B + ( − n B ′ ) β n yielding the complete solution(25) t n = ( A + ( − n A ′ ) α n + ( B + ( − n B ′ ) β n − / with the boundary conditions t = t − = 0 and t = t − , yielding the four relations A + A ′ + B + B ′ = 1 / A − A ′ ) α − + ( B − B ′ ) β − = 1 / A − A ′ ) α + ( B − B ′ ) β = t + 1 / A + A ′ ) α − + ( B + B ′ ) β − = t + 1 / LOSED FORM EQUATIONS FOR TRIANGULAR NUMBERS MULTIPLE OF OTHER TRIANGULAR NUMBERS5 yielding A = α ( α + 1) (cid:0) − β (cid:1) + 2 t ( α − α − β ) A ′ = α ( α − (cid:0) − β (cid:1) − t ( α + 1)4 ( α − β ) B = β ( β + 1) (cid:0) α − (cid:1) − t ( β − α − β ) B ′ = β ( β − (cid:0) α − (cid:1) + 2 t ( β + 1)4 ( α − β ) Closed forms for k = 5 . The smallest case of k with rank two is k = 5 (seeOEIS [13], A077259), for which κ = 8 yielding α = p √ √ and β = p − √ − √ , with t = 2 . The complete solution is then t n = 120 (cid:16)(cid:16)(cid:16) √ (cid:17) − ( − n √ (cid:17) (cid:16) √ (cid:17) n (26) + (cid:16)(cid:16) − √ (cid:17) + ( − n √ (cid:17) (cid:16) − √ (cid:17) n (cid:17) − which yields for even n (27) t n = 120 (cid:16)(cid:16) √ (cid:17) (cid:16) √ (cid:17) n + (cid:16) − √ (cid:17) (cid:16) − √ (cid:17) n (cid:17) − and for odd n (28) t n = 120 (cid:16)(cid:16) √ (cid:17) (cid:16) √ (cid:17) n + (cid:16) − √ (cid:17) (cid:16) − √ (cid:17) n (cid:17) − Closed forms for k = 8 . For k = 8 (see OEIS [13], A336623), r = 2 , for which κ = 16 yielding α = p
17 + 12 √ (cid:0) √ (cid:1) and β = p − √ (cid:0) − √ (cid:1) ,with t = 5 . The complete solution is then t n = 18 (cid:18)(cid:18) (cid:16) √ (cid:17) − ( − n (cid:16) √ (cid:17)(cid:19) (cid:16) √ (cid:17) n (29) + (cid:18) (cid:16) − √ (cid:17) − ( − n (cid:16) − √ (cid:17)(cid:19) (cid:16) − √ (cid:17) n (cid:19) − which can be dissociated for even n , yielding(30) t n = 116 (cid:18)(cid:16) √ (cid:17) (cid:16) √ (cid:17) n + (cid:16) − √ (cid:17) (cid:16) − √ (cid:17) n (cid:19) − and for odd n , yielding(31) t n = 116 (cid:18)(cid:16) √ (cid:17) (cid:16) √ (cid:17) n + (cid:16) − √ (cid:17) (cid:16) − √ (cid:17) n (cid:19) − Similarly, the closed forms for the three other variables are:- for ξ n (3) (see OEIS [13], A336625):(32) ξ n = 14 (cid:18)(cid:16) − ( − n √ (cid:17) (cid:16) √ (cid:17) n +1 + (cid:16) − n √ (cid:17) (cid:16) − √ (cid:17) n +1 (cid:19) − LOSED FORM EQUATIONS FOR TRIANGULAR NUMBERS MULTIPLE OF OTHER TRIANGULAR NUMBERS6 yielding for even n (upper sign) and odd n (lower sign):(33) ξ n = 14 (cid:18)(cid:16) ∓ √ (cid:17) (cid:16) √ (cid:17) n +1 + (cid:16) ± √ (cid:17) (cid:16) − √ (cid:17) n +1 (cid:19) − - for T t n (4) (see OEIS [13], A336624): T t n = 1256 (cid:18)(cid:16) − ( − n √ (cid:17) (cid:16) √ (cid:17) n +1) (34) + (cid:16)
11 + ( − n √ (cid:17) (cid:16) − √ (cid:17) n +1) (cid:19) − yielding for even n (upper sign) and odd n (lower sign):(35) T t n = 1256 (cid:18)(cid:16) ∓ √ (cid:17) (cid:16) √ (cid:17) n +1) + (cid:16) ± √ (cid:17) (cid:16) − √ (cid:17) n +1) (cid:19) − - for T ξ n (5) (see OEIS [13], A336626): T ξ n = 132 (cid:18)(cid:18) (cid:16) √ (cid:17) − ( − n (cid:16) √ (cid:17)(cid:19) (cid:16) √ (cid:17) n (36) + (cid:18) (cid:16) − √ (cid:17) − ( − n (cid:16) − √ (cid:17)(cid:19) (cid:16) − √ (cid:17) n (cid:19) − yielding for even n :(37) T ξ n = 132 (cid:18)(cid:16) √ (cid:17) (cid:16) √ (cid:17) n + (cid:16) − √ (cid:17) (cid:16) − √ (cid:17) n (cid:19) − and for odd n :(38) T ξ n = 132 (cid:18)(cid:16) √ (cid:17) (cid:16) √ (cid:17) n + (cid:16) − √ (cid:17) (cid:16) − √ (cid:17) n (cid:19) − Closed form equations for r = 3 ( ρ = 1 ). The homogeneous solution (12)yields a complete solution that reduces to t n = (cid:18) A + A ′ cos (cid:18) nπ (cid:19) + A ′′ sin (cid:18) nπ (cid:19)(cid:19) α n (39) + (cid:18) B + B ′ cos (cid:18) nπ (cid:19) + B ′′ sin (cid:18) nπ (cid:19)(cid:19) β n − /
2= ( Aα n + Bβ n ) + ( A ′ α n + B ′ β n ) cos (cid:18) nπ (cid:19) + ( A ′′ α n + B ′′ β n ) sin (cid:18) nπ (cid:19) − / LOSED FORM EQUATIONS FOR TRIANGULAR NUMBERS MULTIPLE OF OTHER TRIANGULAR NUMBERS7 with the boundary conditions t = t − = 0 , t = t − and t = t − , giving sixrelations from which one finds the following expressions of the six constants A = α (cid:0) α − β (cid:1) + (2 t + 1) α (cid:0) − β (cid:1) + (2 t + 1) ( α − α − β ) A ′ = α (cid:0) α + β (cid:1) − (2 t + 1) α (cid:0) − β (cid:1) − (2 t + 1) ( α + 2)6 ( α − β ) A ′′ = √ α (cid:0) β + (2 t + 1) α (cid:0) − β (cid:1) − (2 t + 1) (cid:1) α − β ) B = β (cid:0) α − β (cid:1) + (2 t + 1) β (cid:0) α − (cid:1) − (2 t + 1) ( β − α − β ) B ′ = − β (cid:0) α + 2 β (cid:1) − (2 t + 1) β (cid:0) α − (cid:1) + (2 t + 1) ( β + 2)6 ( α − β ) B ′′ = √ β (cid:0) − α + (2 t + 1) β (cid:0) α − (cid:1) + (2 t + 1) (cid:1) α − β ) The smallest case of k with rank three is k = 10 , for which κ = 18 yielding α = (cid:0)
19 + 6 √ (cid:1) = (cid:0) √ (cid:1) and β = (cid:0) − √ (cid:1) = (cid:0) − √ (cid:1) , with t = 1 , t = 6 .2.5. Closed form equations for r = 4 ( ρ = 2 ) . The homogeneous solution (13)yields a complete solution that reduces to t n = (cid:16) A + ( − n A ′ + A ′′ cos (cid:16) nπ (cid:17) + A ′′′ sin (cid:16) nπ (cid:17)(cid:17) α n (40) + (cid:16) B + ( − n B ′ + B ′′ cos (cid:16) nπ (cid:17) + B ′′′ sin (cid:16) nπ (cid:17)(cid:17) β n − / and that simplifies as follows:- for n ≡ mod , t n = ( A + A ′ + A ′′ ) α n + ( B + B ′ + B ′′ ) β n − / - for n ≡ mod , t n = ( A − A ′ + A ′′′ ) α n + ( B − B ′ + B ′′′ ) β n − / - for n ≡ mod , t n = ( A + A ′ − A ′′ ) α n + ( B + B ′ − B ′′ ) β n − / - for n ≡ mod , t n = ( A − A ′ − A ′′′ ) α n + ( B − B ′ − B ′′′ ) β n − / with the boundary conditions t = t − = 0 , t = t − , t = t − and t = t − , givingeight relations yielding the eight constants LOSED FORM EQUATIONS FOR TRIANGULAR NUMBERS MULTIPLE OF OTHER TRIANGULAR NUMBERS8 A = α (cid:0) α − β (cid:1) + (2 t + 1) α (cid:0) α − β (cid:1) + (2 t + 1) α (cid:0) − αβ (cid:1) + (2 t + 1) ( α − α − β ) A ′ = α (cid:0) α + β (cid:1) − (2 t + 1) α (cid:0) α + β (cid:1) + (2 t + 1) α (cid:0) αβ (cid:1) − (2 t + 1) ( α + 1)8 ( α − β ) A ′′ = α + (2 t + 1) α β − (2 t + 1) α − (2 t + 1)4 ( α − β ) A ′′′ = α (cid:0) β + (2 t + 1) α − (2 t + 1) α β − (2 t + 1) (cid:1) α − β ) B = β (cid:0) α − β (cid:1) + (2 t + 1) β (cid:0) α − β (cid:1) + (2 t + 1) β (cid:0) α β − (cid:1) − (2 t + 1) ( β − α − β ) B ′ = − β (cid:0) α + β (cid:1) + (2 t + 1) β (cid:0) α + β (cid:1) − (2 t + 1) β (cid:0) α β + 1 (cid:1) + (2 t + 1) ( β + 1)8 ( α − β ) B ′′ = − β − (2 t + 1) α β + (2 t + 1) β + (2 t + 1)4 ( α − β ) B ′′′ = β (cid:0) − α − (2 t + 1) β + (2 t + 1) α β + (2 t + 1) (cid:1) α − β ) The smallest case of k with rank four is k = 13 , for which κ = 648 yielding α = (cid:0)
649 + 180 √ (cid:1) = (cid:0)
18 + 5 √ (cid:1) and β = (cid:0) − √ (cid:1) = (cid:0) − √ (cid:1) ,with t = 3 , t = 21 and t = 234 .3. Conclusions
We have shown that for triangular numbers multiple of other triangular numbers,closed form equations can be developed instead of recurrent relations, to calculatedirectly the values of triangular numbers and their indices without knowing thewhole preceding sequence of values. We developed the theoretical equations forfour cases of ranks from 1 to 4 and we applied them in several examples for thenon-square multipliers 2, 3, 5 and 8.
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