A256537 -id:A256537 - cp's OEIS frontend

A151723 Total number of ON states after n generations of cellular automaton based on hexagons.

0, 1, 7, 13, 31, 37, 55, 85, 127, 133, 151, 181, 235, 289, 331, 409, 499, 505, 523, 553, 607, 661, 715, 817, 967, 1069, 1111, 1189, 1327, 1489, 1603, 1789, 1975, 1981, 1999, 2029, 2083, 2137, 2191, 2293, 2443, 2545, 2599, 2701, 2875, 3097, 3295

Offset: 0

David Applegate and N. J. A. Sloane, Jun 13 2009

nonn

Analog of A151725, but here we are working on the triangular lattice (or the A_2 lattice) where each hexagonal cell has six neighbors.
A cell is turned ON if exactly one of its six neighbors is ON. An ON cell remains ON forever.
We start with a single ON cell.
It would be nice to find a recurrence for this sequence!
Has a behavior similar to A182840 and possibly to A182632. - Omar E. Pol, Jan 15 2016

S. M. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962 (see Example 6, page 224).

N. J. A. Sloane, Table of n, a(n) for n = 0..4095 [First 1026 terms from David Applegate and N. J. A. Sloane]
David Applegate, The movie version
David Applegate and N. J. A. Sloane, Table of n, A151724(n), A151723(n) for n = 0..1025
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Bradley Klee, Log-periodic coloring, over the half-hexagon tiling.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021

Cf. A147562, A151724, A151725, A161206, A161644, A169779, A169780, A170898, A170905, A182632, A182840, A256536, A256537.

A151723[0] = 0; A151723[n_] := Total[CellularAutomaton[{10926, {2, {{2, 2, 0}, {2, 1, 2}, {0, 2, 2}}}, {1, 1}}, {{{1}}, 0}, {{{n - 1}}}], 2]; Array[A151723, 47, 0](* JungHwan Min, Sep 01 2016 *)
A151723L[n_] := Prepend[Total[#, 2] & /@ CellularAutomaton[{10926, {2, {{2, 2, 0}, {2, 1, 2}, {0, 2, 2}}}, {1, 1}}, {{{1}}, 0}, n - 1], 0]; A151723L[46] (* JungHwan Min, Sep 01 2016 *)

a(n) = 6*A169780(n) - 6*n + 1 (this is simply the definition of A169780).
a(n) = 1 + 6*A169779(n-2), n >= 2. - Omar E. Pol, Mar 19 2015
It appears that a(n) = a(n-2) + 3*(A256537(n) - 1), n >= 3. - Omar E. Pol, Apr 04 2015

Edited by N. J. A. Sloane, Jan 10 2010

A256536 Corner sequence associated with A151723.

Original entry on oeis.org

1, 4, 9, 18, 27, 36, 53, 78, 95, 104, 121, 150, 187, 220, 261, 318, 351, 360, 377, 406, 443, 480, 533, 618, 703, 752, 793, 866, 967, 1060, 1161, 1286, 1351, 1360, 1377, 1406, 1443, 1480, 1533, 1618, 1703, 1756, 1809, 1902, 2035, 2176, 2325, 2522, 2703, 2784, 2825, 2898, 2999, 3108, 3249, 3470, 3723, 3896, 4013, 4186, 4435, 4672, 4909, 5174, 5303

Offset: 1

Omar E. Pol, Apr 01 2015

nonn

Total number of ON cells after n generations in one of the outside corners of an infinite hexagon-shaped structure on hexagonal grid.
For an animation see "The movie version" in Links section.
Partial sums of A256537.
See also the Formula section in A256537.
Compare A256138.

David Applegate, The movie version
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A151723, A153006, A169779, A256138, A256537.

A322663 First differences of A322662 divided by 12.

Original entry on oeis.org

1, 1, 7, 1, 6, 11, 14, 3, 11, 14, 25, 5, 18, 21, 37, 4, 11, 21, 50, 17, 31, 50, 50, 13, 32, 39, 70, 10, 42, 41, 81, 4, 11, 21, 50, 24, 57, 74, 89, 40, 62, 84, 105, 48, 66, 85, 111, 18, 37, 64, 151, 41, 80, 126, 131, 29

Offset: 1

Bradley Klee, Dec 22 2018

nonn

Unlike A322050, this sequence contains only finitely many 1's. However, the Cellular Automaton and its counting sequences still admit a 2^n fractal structure (Cf. A322662). The subsequences L_n = {a(2^n), a(2^n+1), ... a(2^(n+1)-1)} appear to approach a limit sequence L_{oo}, starting with 4 ON cells. Of these 4, one is a "pioneer" at distance d*2^n from the origin, with d the distance of one knight step. The other three of four ON cells are due to retrogressive growth.

			Written as a 2^k triangle:
1,
1, 7,
1, 6,  11, 14,
3, 11, 14, 25, 5,  18, 21, 37,
4, 11, 21, 50, 17, 31, 50, 50, 13, 32, 39, 70,  10, 42, 41, 81,
4, 11, 21, 50, 24, 57, 74, 89, 40, 62, 84, 105, 48, 66, 85, 111, ...

Hexagonal: A151724, A170898, A256537. Square: A147582, A147610, A048883; A319019, A322050, A322049. Lower Bound: A038573.

HexStar=2*Sqrt[3]*{Cos[#*Pi/3+Pi/6],Sin[#*Pi/3+Pi/6]}&/@Range[0,5];
MoveSet2 =Join[2*HexStar+RotateRight[HexStar],2*HexStar+RotateLeft[HexStar]];
Clear@Pts;Pts[0] = {{0, 0}};
Pts[n_]:=Pts[n]=With[{pts=Pts[n-1]},Union[pts,Cases[Tally[Flatten[pts/.{x_,y_}:> Evaluate[{x,y}+#&/@MoveSet2],1]],{x_,1}:>x]]];
Abs[(1/12)*Subtract@@#&/@Partition[Length[Pts[#]]&/@Range[0,32],2,1]]

a(n) = (A322662(n)-A322662(n-1))/12.

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A151723 Total number of ON states after n generations of cellular automaton based on hexagons.

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A256536 Corner sequence associated with A151723.

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A322663 First differences of A322662 divided by 12.

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