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

Author

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

Keywords

• nonn

Comments

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

References

• 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).

Links

• 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

Crossrefs

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

Programs

Mathematica

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 *)

Formula

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

Extensions

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