This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A363760 #39 Dec 14 2023 08:56:43
%S A363760 1,8,9,10,40,72,106,218,256,408,424,872,1178,2336,2522,2952,4712,
%T A363760 10088,13290,26648,28906,33784,53160,115624,150842,303784,330138,
%U A363760 385624,603368,1320552,1716170,3462216,3765322,4397144,6864680,15061288,19543834,39454792,42921274,50118936,78175336,171685096
%N A363760 Cycle lengths obtained by repeated application of Klaus Nagel's strip bijection, as described in A307110.
%C A363760 Description provided by Klaus Nagel: (Start)
%C A363760 The strip bijection s (A307110) maps a point P[i,j] from a Z X Z grid to Q[u,v] taken from a second grid obtained from the first one by a rotation by Pi/4 around the origin. The coordinates [i,j] and [u,v] refer to the respective grids. If [u,v] also are considered as coordinates of the first grid, the mapping [i,j] --> [u,v] establishes a permutation of the grid points of Z X Z.
%C A363760 Cycles of this permutation are evaluated; the sequence shows the occurring cycle lengths L.
%C A363760 Q[u,v] is located close to P[i,j]. Changing the reference to the other grid causes a rotation by Pi/4. Hence after eight permutation steps any point should return to the vicinity of its starting position. (End)
%C A363760 Therefore the provided visualizations also include graphs showing only every 8th point for cycles with L divisible by 8.
%C A363760 Examples of cycles with lengths > 10^9 are L = 2536863994 for the starting position [1761546, 1379978], L = 5574310746 for start [5207814, 6746677], and L = 6508768664 for start [7983336, 8380845].
%H A363760 Hugo Pfoertner, <a href="/A363760/a363760.gif">Animated visualization of cycle with L=256</a>, all visited points shown.
%H A363760 Hugo Pfoertner, <a href="/A363760/a363760_1.gif">Animated visualization of cycle with L=256</a>, every 8th visited point shown.
%H A363760 Hugo Pfoertner, <a href="http://www.randomwalk.de/sequences/a363760_40_53160.pdf">Visualization of all terms from L=40 to L=53160</a>. Zoom in to see details.
%H A363760 Hugo Pfoertner, <a href="https://www.randomwalk.de/sequences/a363760_115624.pdf">Illustration of a(24) = 115624</a>.
%H A363760 Hugo Pfoertner, <a href="https://www.randomwalk.de/sequences/a363760_150842_2.pdf">Illustration of a(25) = 150842</a>.
%H A363760 Hugo Pfoertner, <a href="https://www.randomwalk.de/sequences/a363760_303784.pdf">Illustration of a(26) = 303784</a>.
%H A363760 Hugo Pfoertner, <a href="https://www.randomwalk.de/sequences/a363760_330138_5.pdf">Illustration of a(27) = 330138</a>.
%H A363760 Hugo Pfoertner, <a href="https://www.randomwalk.de/sequences/a363760_385624.pdf">Illustration of a(28) = 385624</a>.
%H A363760 Hugo Pfoertner, <a href="https://www.randomwalk.de/sequences/a363760_603368.pdf">Illustration of a(29) = 603368</a>.
%H A363760 Hugo Pfoertner, <a href="https://www.randomwalk.de/sequences/Cycle_L6508768664.pdf">Illustrations of a cycle of length 6508768664</a>, including zoom images of the self-similar path details, December 2023.
%H A363760 Hugo Pfoertner, <a href="/A363760/a363760.txt">Examples of starting points for all known cycle lengths</a>, July 2023.
%e A363760 a(1) = 1: p(0, 0) -> [0, 0], p(1, 0) -> [1, 0]. Points mapped onto themselves.
%e A363760 a(2) = 8: [0, 1] -> [-1, 1] -> [-2, 0] -> [-1, -1] -> [0, -1] -> [1, -1] -> [2, 0] -> [1, 1] -> [0, 1].
%e A363760 a(3) = 9: [1, 6] -> [-3, 5] -> [-6, 2] -> [-6, -2] -> [-3, -5] -> [1, -6] -> [5, -4] -> [6, 0] -> [5, 4] -> [1, 6].
%e A363760 a(4) = 10: [0, 2] -> [-1, 2] -> [-2, 1] -> [-2, -1] -> [-1, -2] -> [0, -2] -> [1, -2] -> [2, -1] -> [2, 1] -> [1, 2] -> [0, 2].
%e A363760 List of start points and corresponding cycle lengths:
%e A363760 y 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
%e A363760 x \---------------------------------------------------------------
%e A363760 0 | 1 8 10 8 8 40 8 8 8 40 8 8 106 8 8 40 8
%e A363760 1 | 1 8 10 8 8 40 9 40 8 8 106 40 106 8 8 40 8
%e A363760 2 | 8 10 8 8 8 8 8 8 8 8 40 106 8 8 8 8 40
%e A363760 3 | 8 8 8 8 40 9 8 8 8 8 8 8 8 106 8 8 8
%e A363760 4 | 8 8 8 40 8 40 8 8 8 8 8 8 8 8 106 8 8
%e A363760 5 | 8 40 8 40 9 8 8 8 8 8 8 8 8 8 106 8 8
%e A363760 6 | 9 40 9 8 8 40 8 40 106 40 106 8 8 8 106 72 8
%e A363760 7 | 8 8 8 8 8 8 8 8 40 106 8 106 8 106 8 8 72
%e A363760 8 |40 8 8 8 8 8 40 106 8 106 8 8 8 8 8 8 8
%e A363760 9 | 8 8 8 8 8 8 106 40 106 8 8 8 8 8 8 8 8
%e A363760 10 | 8 40 106 8 8 8 8 8 8 8 8 40 8 40 8 8 72
%e A363760 11 |40 106 40 8 8 8 8 106 8 8 40 8 8 8 40 72 8
%e A363760 12 | 8 106 8 106 8 8 8 106 8 8 40 8 8 8 40 8 8
%e A363760 13 | 8 8 8 106 8 8 8 106 8 8 40 8 8 8 40 8 8
%e A363760 14 | 8 8 8 8 106 8 106 8 8 8 8 40 8 40 8 8 8
%e A363760 15 | 8 40 8 8 8 8 8 72 8 8 72 8 8 8 8 8 40
%e A363760 16 | 8 8 40 8 8 8 72 8 8 8 8 72 8 8 8 40 8
%e A363760 .
%e A363760 a(9) = 256: See links to animated visualizations.
%o A363760 (PARI) C=cos(Pi/8); S=sin(Pi/8); T=S/C; \\ Global constants
%o A363760 \\ The mapping function p
%o A363760 \\ PARI's default precision of 38 digits is sufficient up to abs({x,y})<10^17
%o A363760 p(i,j) = {my (gx=i*C-j*S, gy=i*S+j*C,k, xm, ym, v=[0,0]); k=round(gy/C); ym=C*k;xm=gx+(gy-ym)*T; v[1]=round((xm-ym*T)*C); v[2]=round((ym+v[1]*S)/C); v};
%o A363760 \\ cycle length
%o A363760 cycle(v) = {my (n=1, w=p(v[1],v[2])); while (w!=v, n++; w=p(w[1],w[2])); n};
%o A363760 a363760 (rmax) = {my (L=List()); for (x=0, rmax, for(y=x, rmax, my(c=cycle([x,y])); if(setsearch(L,c)==0, listput(L,c); listsort(L,1)))); L};
%o A363760 a363760(500) \\ takes a few minutes, terms up to a(19), check completeness of list with larger rmax
%Y A363760 Cf. A362955, A362956, A367146, A367893.
%Y A363760 Cf. A367148 (analog of this sequence, but for the triangular lattice).
%K A363760 nonn
%O A363760 1,2
%A A363760 _Hugo Pfoertner_, Jun 26 2023