The Pagoda Sequence: a Ramble through Linear Complexity, Number Walls, D0L Sequences, Finite State Automata, and Aperiodic Tilings
TT. Neary, D. Woods, A.K. Seda and N. Murphy (Eds.):The Complexity of Simple Programs 2008.EPTCS 1, 2009, pp. 130–148, doi:10.4204/EPTCS.1.13 c (cid:13)
F. Lunnon
The Pagoda Sequence: a Ramble through Linear Complexity,Number Walls, D0L Sequences, Finite State Automata, andAperiodic Tilings
Fred Lunnon
National University of Ireland Maynooth, Co. Kildare, Ireland
We review the concept of the number wall as an alternative to the traditional linear complexity profile(LCP), and sketch the relationship to other topics such as linear feedback shift-register (LFSR) andcontext-free Lindenmayer (D0L) sequences. A remarkable ternary analogue of the Thue-Morse se-quence is introduced having deficiency 2 modulo 3, and this property verified via the re-interpretationof the number wall as an aperiodic plane tiling.
In the early 1970’s the availability of the Berlekamp-Massey algorithm led to the emergence of theLinear Complexity Profile (LCP), as a measure of how well a sequence of (say) binary digits could beapproximated by a Linear Feedback Shift-Register (LFSR) — a topic of some practical importance inthe design of cryptographic key-stream sequences.A less established alternative, previously known to rational approximation specialists by the some-what unimaginative term
C-table , is the number wall — an array of Hankel determinants formed fromconsecutive intervals of the sequence — which lends itself better to geometrical interpretation than thetraditional LCP.An algorithm for number-wall computation, generalising the classical Jacobi recurrence to the previ-ously intractable case of zero determinants, was later discovered by the author, who typically then failedto get around to actually publishing it for another 25 years. It is applicable to sequences over any integraldomain, and with care can be implemented to cost constant time per entry computed.A particular area of interest involves sequences whose complexity according to this model is in someway extreme, such as that proposed by Rueppel with a so-called ‘perfect’ LCP. Sequences with ‘perfect’number-walls are harder to find, in fact over a finite domain they appear not to be possible: a probabilisticargument gives approximate bounds on the depth of such tables, confirmed by computer searches modulo2 and 5.Despite this in 1997 was discovered a remarkable sequence with modulo 3 deficiency 2, that is itsternary number-wall contains only isolated zeros — or in plainer language, no linear recurrence or LFSRof order m spans any 2 m + p = k − Pagoda sequence resembles that of the classical square-free Thue-Morseternary sequence: an auxiliary sequence is generated via a D0L system, then mapped to the target se-quence via a final extension morphism. Such D0LEC (D0L with extension and constant width) or ‘auto-matic’ sequences have some claim to form a natural complexity class immediately above the LFSR class,combining greater flexibility with accessible distribution properties. a r X i v : . [ c s . D M ] J un . Lunnon A sequence [ S n ] is a linear recurring or linear feedback shift register (LFSR) sequence of order r , whenthere exists a nonzero vector [ J i ] (the relation ) of length r + r ∑ i = J i S n + i = n .If the relation has been established only for a ≤ n ≤ b − r we say that the relation spans S a , . . . , S b , with a = − ∞ and b = + ∞ permitted.Sequences may have as elements members of any integral domain: in applications the domain willusually be the integers or some prime (often binary) finite field. LFSR sequences over finite fields arediscussed comprehensively in [5] § (Shifted) Linear Complexity Profile (LCP/SLCP) represented an attempt to establish a relevant quantitiveformalism: given [ S n ] , its LCP is an auxiliary sequence with m -th term the order of the minimal LFSRspanning segment S , . . . , S m − ; the SLCP generalises this reluctantly into two dimensions by consideringthe order of S n , . . . , S n + m − , where both m and n vary.In recent years linear complexity has made little progress; and it is my contention that the majorculprit is the accidental manner in which LCP’s were contrived. The Berlekamp-Massey algorithmhad recently been developed, providing a means of computing the minimal relation spanning n termsof a sequence in time quadratic in n . This seems then to have been seized upon by both coding andcomplexity communities — the latter simply discarding the components [ J i ] of the relation, retainingonly the order r .To introduce a personal note at this point, I have to confess to having never felt comfortable withBerlekamp-Massey: its application is tricky — for instance, the intermediate vectors it generates cannotbe relied upon to represent relations spanning a prefix of the segment — and its proof (see [5]) strikesme as both complicated and lacking obvious direction.A more natural and elementary alternative considers instead the simultaneous linear equations for therelation components [ J i ] in terms of the sequence elements [ S n ] . Easily, these have a solution just whenthe Toeplitz determinant [or with an extra reflection,
Hankel or persymmetric ] S mn = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S n S n + . . . S n + m S n − S n . . . S n + m − ... ... . . . ... S n − m S n − m + . . . S n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) vanishes.A zero entry S mn indicates a relation of order r ≤ m spanning the segment [ S n − m , . . . , S n + m ] . If thesequence is in fact generated by a single LFSR of order r , the table will be zero from row r onwards:32 The Pagoda Sequence therefore this number-wall bears the same relation to an LFSR sequence as does the difference table to apolynomial sequence (where S n is a polynomial function of n ); in fact, one generalises the other, to theextent that every polynomial sequence of degree r − r , with relation given bythe vanishing of its r -th difference.These determinants can also be computed in quadratic time, via an algorithm not only progressive [sothe time becomes effectively linear for a table of many values], but beguilingly simple and symmetrical,and classical — being a special case of a well-known pivotal condensation rule or extensional identitycredited variously to Sylvester, Jacobi, Desnanot, Dodgson, Frobenius: S m , n = S m + , n S m − , n + S m , n + S m , n − . Unfortunately, formulating a corresponding recursive algorithm, expressing each row in terms of thetwo previous, S − , n = , S − , n = , S , n = S n , S m , n = (cid:0) S m − , n − S m − , n + S m − , n − ) (cid:1) / S m − , n for m > S m − , n = Pad´e block theorem (see [4]) to Pad´e table specialists [who have incidentally been collectively responsible for a remarkablenumber of bogus proofs of it] is that zero entries occur only as continuous square regions, surrounded byan inner frame of nonzeros [easily seen by the Sylvester identity to comprise a geometric sequence alongeach edge].Some time around 1975, I succeeded in generalising the recursion to bypass such zero entries. Ironi-cally (given my original motivation) even the statement of these frame theorems demands sufficient pre-liminary background to necessitate relegation to appendix A; and their convoluted and technical proofrequired several attempts, finally involving a combination of methods from ring theory, analysis andalgebraic geometry, and sustained over a period of more than a quarter of a century [7].[John Conway, who took an early interest in this topic, christened the zero regions windows , and thetable a wall of numbers , [2]. Apparently, on first encountering these results, he transcribed them for safekeeping onto his bathroom wall (the way one does); but having moved house by the time the book cameto be written, was obliged to rely on memory, and as a result (to his evident embarrassment) committedtwo separate typographical errors in restating them.]There is plainly a close relationship between SLCP’s and number walls — see [9] for example.However, the more symmetrical definition of the latter considerably facilitates the deployment of gen-uinely two-dimensional geometry in their investigation, as we shall see later; in contrast, the (diagonal,one-dimensional) generating function technique — encouraged by the LCP paradigm — is for exampleunable to probe the central diamond of a number wall at all.
A deterministic context-free Lindenmayer (D0L) system is defined to be a substitution system wherethere is only one production for each symbol; all productions are applied simultaneously; and productionis iterated, starting from some distinguished (stable) symbol, so generating an infinite sequence. . Lunnon k -uniform morphism’ in sect. 6 of [1].]Why should D0L (and particularly D0LEC) systems be worthy of study? LFSR systems arise natu-rally in a number of applications (signal-processing, cryptography), and the number wall is a natural toolwith which to investigate them. When we come to study number walls in turn, their extremal behaviouris observed to occur for D0LEC sequences (which incidentally arise in other unrelated applications aswell). So D0LEC sequences in some sense constitute a natural third layer in a complexity hierarchycommencing thus: polynomial sequences, LFSR sequences, D0LEC sequences, . . . The distribution properties of these sequences can easily be established, using classical Markov-process methods [3]. Another bonus is algorithmic: the D0LEC paradigm permits both the computationof a distant term S n of a sequence, and furthermore the inversion of this process to recover n from S n (where this is single-valued), in time of order log n by means of a finite-state automaton — see [1].In illustration of these ideas, we turn now to consider the Thue-Morse sequence. [This is conven-tionally constructed as the fixed point of the morphism 0 → , →
10; however, the following indirectconstruction proves more illuminating.] Recall that a sequence of symbols is square-free when no factorword (of consecutive symbols) is followed immediately by a copy of itself; similarly, a sequence may be cube-free , power-free .Consider the D0L system on 4-symbols defined by the generating morphism Φ : A → BC , B → BD , C → CA , D → CB ;notice the symmetry of Φ under the permutation ( AD )( BC ) . Starting from B and applying Φ repeatedlygives what turns out to be a square-free right-infinite quaternary sequence: [ V n ] = BDCBCABD CABCBDCB CABCBDCA BDCBCABD . . . [This could be made left- and right-infinite by starting with AB or CB and fixing the origin in the centre;but then Φ rather than Φ would be required to obtain stability.]The final morphism A → , B → , C → , D → , now yields the classical cube-free binary Thue-Morse sequence [ T n ] = . . . , explicitly T n equals the sum modulo 2 of the digits of n when expressed in binary. Alternatively, the finalmorphism A → , B → , C → , D → , yields the related ternary sequence [ U n ] = . . . , which can be shown to be square-free. Proofs are given in appendix B; they bear comparison with rathercomplicated ad-hoc arguments available elsewhere, e.g. [6].34 The Pagoda Sequence
More significantly, other final morphisms may be tailored to produce new sequences, such as A → , B → , C → , D → , yielding a binary sequence which has no squared words of length exceeding 6:01001001 10110100 10110110 01001001 . . . , and A → , B → , C → , D → , with no squares exceeding length 4 (optimal):01110010 10000111 10001101 01110010 10001101 01111000 . . . We propose to illustrate the discussion using an interactive Java application which displays number wallsof various special sequences modulo a given prime. Entries are encoded as coloured pixels: white for 0,black for 1, grey for 2; or red for 2, green for 3, blue for 4, etc. interactively; the sequence runs alongtwo rows from the top edge. Program source
ScrollWall.java is available from the author;The implementation is based on the frame theorems (appendix A), incorporating an enhancementto obviate searching when circumnavigating a large window. [The binary case is particularly simple,to the extent that an exceptionally efficient implementation is feasible in the form of a 44-state cellularautomaton based on the Firing-Squad Synchronisation Problem (FSSP) — see [8]; [7] sect.7.] The givenfinite segment must be extended into a periodic sequence, to avoid algorithmic complications resultingfrom the presence of a boundary: therefore in general, only the triangular north quarter of a (square)graphical display is significant; although in special cases, intelligent choice of segment length n mayimprove this situation. Since a sequence with period r is LFSR with order at most r , the number ofnonzero rows (including the initial row of empty determinants) for a segment of length n columns mustbe at most n + q elements, it can be shown that for a random sequence, the asymptoticmean density of size- g (or g × g ) windows exists (in some suitably weak sense), and equals ( q − ) / ( q + ) q · / q g ;for example, Fig.1 , Fig.2 modulo q = , . It is tempting to employ this result as a test for randomness: for example, counting the numbersof windows of each size in a suitably large portion of the wall, then applying the χ test to the fre-quencies. A discouraging counterexample is the sum of the Thue-Morse and Rook sequences modulo 2(see Fig.3 ), which passes this test with flying colours, despite being generated by the 8-symbol D0LECsystem: A → Ab , B → Ad , C → Cb , D → Cd , a → aB , b → aD , c → cB , d → cD ; A → , B → , C → , D → , a → , b → , c → , d → . Our major target in this essay is the investigation of extremal walls: by which is meant, the extentto which a number wall may deviate from typical window distribution. One pretext for this activity . Lunnon
The Pagoda Sequence
Figure 2: Libran3 . Lunnon
The Pagoda Sequence
Figure 4: Rueppelis exposure of the limitations of the paradigm; but it might be more honest to prefer the serendipitousjustification, that some of the graphic art so produced is simply rather striking [and might become moreso, were the author’s casually primitive palette to be refined!]The (not overly impressive) example of the Rueppel sequence makes a point about the limitations ofthe original LCP concept. Its definition is S n = (cid:26) n = k − k ;0 otherwise.It was proposed as an example of a binary sequence having ‘perfect’ LCP, which in number-wall termsimplies a continuous nonzero diagonal staggering from one corner to the opposite. Elsewhere though,its wall is perfectly appalling, composed almost exclusively of windows increasing exponentially in size(see Fig.4 ).But it suggests an analogous though considerably tougher challenge, which we proceed to take up:to determine the extent to which a (binary, say) wall can avoid zero entries. A combinatorial argumentbased on the frame theorems shows easily that any extended region of the wall has local zero-densityat least 1 / Rook sequence [ R n ] as the digitpreceding the least-significant 1 in the binary expansion of n , or 0 if n = n = = R = [ R n ] is a binary sequence, and a recursion for it is R − n = − R n for n (cid:54) = R n = R n ; R n + = n mod 2 . . Lunnon n ≥ [ R n ] = . . . Finally, define the
Knight sequence by K n = R n + − R n − ( mod 2 ) . For ternary walls, the situation is rather similar: an essentially unique nonzero local pattern exists,composed of alternating zigzag stripes of + − A → ACB , B → BCB , C → EDF , D → DDD , E → EDD , F → DDF ; A → , B → , C → , D → , E → , F → . Starting from A , the first few terms of generated and final sequences are ACBEDFBCB EDDDDDDDF BCBEDFBCB . . . ; [ Z n ] = . . . Accepting that a total absence of zeros (on rows m ≥ −
1) is not possible, we can instead attemptin various ways to circumscribe their occurrence. Rather than become involved in somewhat reconditequestions regarding what exactly might be meant by the term density in this context, we shall considerthe more concrete problem of bounding the size of the windows.40
The Pagoda Sequence
Figure 6: ZigZagFigure 7: Def2Mod2A simple probabilistic argument can be mounted suggesting that, when the domain is a finite fieldwith q elements, the size of the maximum window occurring within the first m rows of a wall will be ofthe order of log q m ; and more strongly, that the probability of a sequence having no windows larger thanthis bound is zero. [This contrasts with the situation for square-free sequences, where the correspondingprobability is nonzero for q > d or greater, for small values of the [in LCP jargon] deficiency d . Thisendeavour is highly speculative: first the critical depth m must be established such that no satisfactorysequence exists with greater depth; then a sufficiently long segment constructed for an evident period tobecome established.The resulting handful of sequences is shown in the table: all are periodic with period t , and the order r equals the final depth m satisfying the deficiency bound — that is, as soon as the bound fails, the entirewall vanishes — and m seems to increase exponentially with d as expected. Confidence in these resultsis encouraged by the presence of adventitious symmetries, such as the 0-1 alternating subsequence at oddpositions of case d =
4. See Fig.7 , Fig.8 , Fig.9 .Now what about ternary walls? Deficiency d = m = r =
2. But when our search program is let loose on d =
2, the first of a number ofstrange things happens — or in this case, fails to happen — the depth goes on increasing indefinitely,while (necessarily) no period ever properly quite stabilises. To cut quite a long story short, the objectwhich eventually emerges is a remarkably simple D0LEC, has deficiency-2 to any depth we care to . Lunnon d m r t period1 1 1 1 [1]2 5 5 6 [111010]3 19 19 20 [1111010100 1111010010]4 56 56 60 [0001100100 0110110011 00011011001110110001 1001001100 1110010011]5 95+ ? ? (none detected in 800 terms)42
The Pagoda Sequence
Figure 10: Pagodaexamine, and turns out to be essentially identical to the Knight [ K n ] — seen earlier in an unrelatedcontext!To be precise, with R n the Rook sequence as above, the Pagoda sequence is defined by P n = R n + − R n − ( mod 3 ) . The ternary number wall is shown at Fig.10 ; the symmetrical, fractal-like filigree structures for whichit was christened are more easily appreciated after rotation through a quarter-turn, the sequence runningdown the left side.Examination of substantial portions of the number-walls of this sequence modulo p = , , , , , , , , , , , any prime p = k −
1; modulo p =
83 however, this elegant simplicity is confounded by the presence of numerous windows of size 2, to-gether with what appears to be a splendidly lone specimen of size 3 commencing at . Lunnon m = , n =
188 [a specimen discovered only during protracted investigation of an apparent com-piler bug causing ScrollWall to report spurious runtime errors].The Pagoda was not the first, nor the last of its kind to be discovered: but all these, along withthe Knight and Rook sequences, are closely intertwined, in a manner notably reminiscent of our earlieranalysis of the Thue-Morse family. Consider the 4-symbol D0L system A → AB , B → AD , C → CB , D → CD ;applied to A this generates [ V n ] = ABADABCD ABADCBCD ABADABCD CBADCBCD . . . [which can be made infinite both ways by starting instead from DA and choosing the origin to be the firstsymbol of the (inflated) original A .] Applying the final morphism A → , B → , C → , D → [ P n ] , for n ≥ n ]; A → , B → , C → , D → [ K n ] .Other deficiency-2 variations on the Pagoda may be concocted by varying the final morphism. Alsonotice that the generator is not symmetric under either transposition ( AC ) or ( BD ) : so these provide aset of 4 distinct generators, each of which could be used to yield an alternative quaternary sequence.Applying the final morphism A → , B → , C → , D → [ R n ] .Modulo these variations, and the continuum of variants obtained by shifting the origin repeatedlyduring generation, it seems quite plausible that the Pagoda is the unique ternary sequence with thisdeficiency. At this point, we have some probabilistic arguments and experimental evidence to support the conjecturesthat: If p mod 4 = p =
2, then the maximum depth m to which deficiency d can be maintained byany number wall modulo p is finite, bounded by order log p d ;If p mod 4 = −
1, then the number wall modulo p of the Pagoda sequence has bounded deficiency(dependent only on p ) to any depth; in particular, for p = d = p , or over the integers: while there is some numerical structure visible here which might forma basis for an inductive construction, overall this prospect is not promising.A more unexpected route proves at least partially successful: invoking a two-dimensional geomet-rical version of the D0LEC paradigm, extending the representation of the sequence via [ V n ] above, into44 The Pagoda Sequence one of the entire wall as a plane quasi-crystallographic tiling. In part this is suggested by close visualinspection of the diagram, which reveals (at the cost of substantial hazard to eyesight) that the ‘pago-das’ recurring at various scales throughout the wall are embedded in repetitive diamonds , square regionsrotated through a one-eighth turn.Factors to be taken into account in the formalisation of this concept include:Interaction between faces, edges and vertices of tiles;Non-trivial point symmetries tiles may possess;Choice of an appropriate translation of tiling origin;Extent to which tiles are open or closed subsets of the plane;Determination of tile size, or D0L extension width;Determination of number of distinct tiles, or D0L symbol count.All these factors, along with other details relevant to implementation only at a detailed level, need betaken into account in the design of a program to (as it were) tile a wall — to specify the precise spatial‘inflation’ morphism generating it, along with the extension ‘pattern’ on each tile.It is natural to align the vertices of a tile with entries of the number wall, so that an entry at a vertex isshared between 4 adjacent tiles, at an edge between 2. This presents a conflict between notational clarityand computational simplicity, resolved by including the entire boundary in tile morphism diagrams (ap-pendix C); while to actually apply a morphism, the boundary must be shrunk and displaced by a half-unitalong each axis, so that a tile comprises only complete entries.In order to verify the frame relations between wall entries, as well as to keep track of inflation ofvertices and edges along with faces of tiles, the search program actually operates a 4-fold covering ofthe plane by overlapping supertiles having twice the diameter of the faces. A post-processor extractsthe individual inflations of faces etc, possibly resulting in tile extents becoming reducible to smallerdiameter. At this stage also, point-group symmetries of ‘fixed’ tiles are extracted; the number of ‘free’tiles remaining is then substantially reduced.For the ternary Pagoda, the program successfully finds a tiling comprising:Generator inflation diameter 2 (4 subtiles per inflation);Point symmetry group of order 16;Tiling origin at S − , ;Extent diameter of face 4 (partially spanning 25 wall entries);Fixed face count 107, reducing to 13 free;Every free face occurring within distance 35 from the origin;Free vertex count 39, all within distance 165;The full morphism will be found in appendix C. Point symmetries comprise products of vertical reflec-tion, horizontal reflection, complementation of odd rows, complementation of odd columns.Apart from two restricted to meeting the upper zero half-plane m ≤ −
2, every tile has only isolatedzeros: this completes the proof that the deficiency of the ternary Pagoda equals 2.But of course, the existence of this tiling permits us to investigate the wall in much greater detail.For instance, by selectively expanding the D0LEC, any given entry S mn can now be computed in time oforder logarithmic in the distance | m | + | n | from the origin.Again, the deficiency theorem may be considerably sharpened: . Lunnon S mn = S n = P n , then the power of 2 dividing m + n .In particular, no zeros can occur on rows with m odd, nor on column n = m ≥ −
1, that is).Again, applying Markov process analysis to the D0LEC, a 13 ×
13 matrix eigenvalue computationestablishes thatZero entries in this wall possess asymptotic density in a strong sense, and this density equals 3 / ≤ S n = T n + R n ( mod 2 ) mentioned earlier, with window size bounded apparently by order ( log m ) . A Statement of the Frame Theorems.
A zero entry S m , n = window , that is a square g × g zero regionsurrounded by a nonzero inner frame. The nullity of (the matrix corresponding to) a zero entry equals itsdistance h from the (nearest) inner frame edge.The adjacent diagram illustrates a typical window, together with notation employed subsequently: E E E . . . E k . . . E g E g + F B , A A A . . . A k . . . A g A , C g + G g + F B . . . . . . C g G g F B . . . ( P ) → ... ... ...... ... ... ( Q ) . . . ↑ C k G k F k B k ↓ . . . ( R ) ... ... ...... ... ... ← ( T ) . . . C G F g B g . . . . . . C G F g + B , D g + D g . . . D k . . . D D D , C G H g + H g . . . H k . . . H H H The inner frame of a g × g window comprises four geometric sequences, along North, West, East,South edges, with ratios P , Q , R , T resp., and origins at the NW and SE corners. The ratios satisfy PT / QR = ( − ) g ;and the corresponding inner frame sequences A k , B k , C k , D k satisfy A k D k / B k C k = ( − ) gk for 0 ≤ k ≤ g + . The outer frame sequences E k , F k , G k , H k lie immediately outside the corresponding inner, and arealigned with them. They satisfy the relation: For g ≥
0, 0 ≤ k ≤ g + QE k / A k + ( − ) k PF k / B k = RH k / D k + ( − ) k T G k / C k . Proofs are expounded in [7] sect. 3–4.46
The Pagoda Sequence
B Proofs that [ V n ] , [ T n ] , [ U n ] are power-free. We sketch the proofs that these sequences are power-free as claimed. Suppose that [ V n ] is not square-free,and let the earliest occurrence of its shortest non-empty square start at V n for n ≥
0, with length 2 l > l is even: if n is odd, by inspection of Φ there is only one possible value for V n − = V n + l − given V n = V n + l , so there is an equally short square earlier; if n is even, we can apply Φ − to produce a shortersquare of length l /
2. Suppose on the other hand l is odd: then for each i one of V n + i and V n + l + i has aneven subscript, so by inspection has to be B or C . No new pairs are generated after Φ B , so all words oflength 4 occur within Φ B ; the longest composed of B and C only is seen to have length 3. So 2 l ≤ BB or CC , which do not occur in Φ B . By contradiction, [ V n ] is square-free.The inverse morphism from [ U n ] to [ V n − ] is uniquely defined for n ≥
1, given either of the symbols U n ± adjacent to U n : it is described by the schema ( ) ( ) → A , → B , → C , ( ) ( ) → D , where U n ± is parenthesised. If [ U n ] had a square factor with l >
2, its inverse image would also be asquare in [ V n ] , since A and D in corresponding positions necessarily have an adjacent B and C ; but [ V n ] issquare-free. If l = AD or DA , but neither occurs in [ V n ] .The inverse morphism from [ T n ] to [ V n ] is uniquely defined for n ≥
2, given T n + ; it is described bythe schema 0 ( ) → A , ( ) → B , ( ) → C , ( ) → D , where T n + is parenthesised. If [ T n ] had a cubic factor, its inverse image would also be a cube in [ V n ] ,except possibly for the final symbol; but [ V n ] is square-free. C Pagoda Tiling Morphisms
Free tiles are numbered 1–13. The ‘gene’ field diagrams the 2 × × code transform A identityB reflection along rowsC reflection along colsD half-turn rotationI identityJ complement odd rowsK complement odd colsL complement odd rows & cols
02 0 0 0Tile 1: gene 1B 1 , extn 0 0 0 0 0, symm AI,BI;4 1 1 11 . Lunnon
02 0 0 0Tile 2: gene 2 2 , extn 0 0 0 0 0, symm full;2 0 0 0003 1 1 1Tile 3: gene 5 7 , extn 1 2 2 0 1; symm AI;6 2 1 1103B 1 1 1Tile 4: gene 5D 7BJ , extn 1 1 2 0 1; symm AI;8 2 1 11110 1 1 2Tile 5: gene 9 9BK , extn 1 0 2 0 1, symm AI,BK;11 1 1 2116C 2 1 1Tile 6: gene 7BK 7 , extn 1 0 2 0 1, symm AI,BK;10BJ 2 1 1113CJ 1 1 1Tile 7: gene 12 12BL, extn 1 2 0 1 1, symm AI,CI;3J 1 1 1116D 1 1 2Tile 8: gene 5D 5D , extn 1 1 2 2 1, symm AI,BK;11C 2 2 1114D 1 1 1Tile 9: gene 13 13BL, extn 1 1 0 2 1, symm AI,CI;4B 1 1 11 The Pagoda Sequence
18C 2 1 1Tile 10: gene 5B 5B , extn 1 1 2 2 1, symm AI,BK;10B 1 2 21111J 1 1 2Tile 11: gene 7J 7BL , extn 1 0 2 0 1, symm AI,BK;8B 1 1 2118BJ 1 1 2Tile 12: gene 9BJ 12 , extn 1 0 2 1 0, symm AI,CI;8DJ 1 1 2116J 2 1 1Tile 13: gene 9BL 12J , extn 1 0 2 1 0, symm AI,CI;6CJ 2 1 11
References [1] Allouche, Jean-Paul & Shallit, Jeffrey
Automatic Sequences
Cambridge (2003).[2] Conway, J. H. & Guy, R. K.
The Book of Numbers
Springer (1996).[3] Feller, William
An Introduction to Probability Theory and its Applications vol I Wiley (1957).[4] Gragg, W. B.
The Pad´e Table and its Relation to Certain Algorithms of Numerical Analysis
SIAM Review (1972) 1–62.[5] Lidl, R. & Niederreiter, H. Introduction to Finite Fields and their Applications
Cambridge (1997).[6] Lothaire, M.
Combinatorics on Words
Addison-Wesley (1983).[7] Lunnon, W. F.
The Number-Wall Algorithm: an LFSR Cookbook
Article 01.1.1 Journal of Integer Sequences (2001).[8] Minsky, M. Computation: Finite and Infinite Machines
Prentice-Hall (1967).[9] Stephens, N. M.
The Zero-square Algorithm for Computing Linear Complexity Profiles in Mitchell, Chris(ed.)