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 A367148 #15 Apr 21 2024 14:28:55
%S A367148 1,10,12,36,37,56,60,72,84,110,120,154,156,168,192,278,370,398,444,
%T A367148 492,516,564,600,614,660,924,961,1114,1128,1164,1500,1574,1668,1786,
%U A367148 2052,2076,2100,2220,2336,2388,2604,2952,3300,3456,3612,3684,3924,4548,4692,4882,4968
%N A367148 Cycle lengths obtained by repeated application of the strip bijection for the triangular lattice described in A367147.
%C A367148 The repeated application of the bijection function Q described in A367147, which maps a pair of triangular coordinates [i,j] to an image point [m,n], returns to the starting point after a number of steps dependent on the starting point. One mapping step leads to a location that approximately corresponds to a rotation of Pi/6, so that often, but not always, the lengths of the orbits created are multiples of 12. The situation is very similar to that described in the comment to A363760 for the analogous process applied to the square grid. As the lengths of the cycles increase, remarkable self-similar structures emerge; see the visualization of a cycle with a length L > 6*10^8.
%H A367148 Klaus Nagel, <a href="/A367148/a367148.png">Illustration of cycle lengths</a>; points that belong to cycles of the same length are shown with the same color. Randomly selected region in grid.
%H A367148 Hugo Pfoertner, <a href="/A367148/a367148_1.pdf">Visualization of orbits with L <= 1500</a>.
%H A367148 Hugo Pfoertner, <a href="/A367148/a367148.pdf">Orbit of length 617818092</a>.
%e A367148 a(1) = 1: Starting point [0, 0] trivially mapped to [0, 0]; Q([0, 0]) -> [0, 0], Q([1, 0]) -> [1, 0]. Points exactly mapped to rotated location.
%e A367148 a(2) = 10: [2,0] -> [3,-2] -> [2,-3] -> [1,-3] -> [-1,-2] -> [-2,0] -> [-3,2] -> [-2,3] -> [-1,3] -> [1,2] -> [2, 0];
%e A367148 a(3) = 12: [3,0] -> [4,-2] -> [4,-4] -> [2,-5] -> [-1,-4] -> [-3,-2] -> [-4,0] -> [-5,2] -> [-5, 4] -> [-3,5] -> [0,4] -> [2,2] -> [3,0].
%e A367148 .
%e A367148 List of triangular coordinates [i, j] of start points and corresponding cycle lengths:
%e A367148 .
%e A367148 j 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
%e A367148 i \-------------------------------------------------------------------
%e A367148 0 | 1 1 10 10 12 12 56 12 110 12 12 12 12 278 12 12 12
%e A367148 1 | 1 10 10 12 12 12 56 12 12 110 12 12 37 278 12 12 278
%e A367148 2 | 10 10 12 12 56 56 12 110 12 110 37 278 278 12 278 12 278
%e A367148 3 | 12 12 12 12 12 56 12 110 12 110 37 12 278 278 278 12 60
%e A367148 4 | 12 12 12 56 12 110 12 110 12 37 278 12 278 12 60 12 12
%e A367148 5 | 12 56 56 56 12 110 12 12 110 37 278 12 278 12 12 60 12
%e A367148 6 | 12 56 12 110 12 12 12 37 278 278 278 278 12 60 12 60 12
%e A367148 7 | 12 110 12 12 110 12 12 12 278 12 12 278 12 60 12 12 60
%e A367148 8 | 12 12 110 12 110 37 278 37 12 12 12 278 12 12 60 12 398
%e A367148 9 |110 12 110 12 110 37 278 278 12 12 12 12 278 12 398 12 398
%e A367148 10 | 12 12 110 37 12 278 12 278 12 12 278 12 398 398 398 12 12
%e A367148 11 | 12 37 37 278 12 278 278 12 278 12 278 12 398 12 12 12 12
%e A367148 12 | 12 278 278 278 12 12 278 12 278 12 398 12 12 12 12 12 72
%e A367148 13 | 37 278 12 278 278 12 60 12 60 12 398 398 12 12 72 36 72
%e A367148 14 | 12 12 278 12 60 12 60 12 12 398 12 12 12 36 36 12 12
%e A367148 15 | 12 12 278 12 60 12 12 60 12 398 12 12 72 72 12 12 12
%e A367148 16 | 12 12 278 12 12 60 12 60 12 398 12 12 398 72 12 12 72
%o A367148 (PARI) \\ uses mapping function Q defined in PARI program of A367147
%o A367148 cycle(v) = {my (n=1, w=Q(v)); while (w!=v, n++; w=Q(w)); n};
%o A367148 L = List(); \\ global list to support repeated calls of function a367148
%o A367148 a367148(x10min=2, x10max=3, nrep=10000) = {for (n10=x10min, x10max, my (rmax=10^n10); for (n=1, nrep, my (x=random(rmax), y=random(rmax), c=cycle([x, y])); if(setsearch(L, c)==0, print1([c,x,y],", "); listput(L, c); listsort(L, 1)))); L};
%o A367148 \\ De-activate print to avoid output of starting points
%o A367148 a367148(2,3) \\ usually sufficient to get all terms <= 1500, repeat and increase nrep for confirmation; no shortcut for efficient systematic selection of starting points is known.
%Y A367148 Cf. A003136, A307110, A363760, A367147, A367149.
%K A367148 nonn
%O A367148 1,2
%A A367148 _Hugo Pfoertner_, Nov 11 2023