cp's OEIS Frontend

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

%I A151723 #73 Mar 23 2021 10:41:45
%S A151723 0,1,7,13,31,37,55,85,127,133,151,181,235,289,331,409,499,505,523,553,
%T A151723 607,661,715,817,967,1069,1111,1189,1327,1489,1603,1789,1975,1981,
%U A151723 1999,2029,2083,2137,2191,2293,2443,2545,2599,2701,2875,3097,3295
%N A151723 Total number of ON states after n generations of cellular automaton based on hexagons.
%C A151723 Analog of A151725, but here we are working on the triangular lattice (or the A_2 lattice) where each hexagonal cell has six neighbors.
%C A151723 A cell is turned ON if exactly one of its six neighbors is ON. An ON cell remains ON forever.
%C A151723 We start with a single ON cell.
%C A151723 It would be nice to find a recurrence for this sequence!
%C A151723 Has a behavior similar to A182840 and possibly to A182632. - _Omar E. Pol_, Jan 15 2016
%D A151723 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).
%H A151723 N. J. A. Sloane, <a href="/A151723/b151723.txt">Table of n, a(n) for n = 0..4095</a> [First 1026 terms from David Applegate and N. J. A. Sloane]
%H A151723 David Applegate, <a href="/A139250/a139250.anim.html">The movie version</a>
%H A151723 David Applegate and N. J. A. Sloane, <a href="/A151723/a151723.txt">Table of n, A151724(n), A151723(n) for n = 0..1025</a>
%H A151723 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, 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.]
%H A151723 Bradley Klee, <a href="/A151723/a151723.png">Log-periodic coloring, over the half-hexagon tiling</a>.
%H A151723 N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H A151723 N. J. A. Sloane, <a href="https://www.youtube.com/watch?v=9ogbsh8KuEM">Exciting Number Sequences</a> (video of talk), Mar 05 2021
%F A151723 a(n) = 6*A169780(n) - 6*n + 1 (this is simply the definition of A169780).
%F A151723 a(n) = 1 + 6*A169779(n-2), n >= 2. - _Omar E. Pol_, Mar 19 2015
%F A151723 It appears that a(n) = a(n-2) + 3*(A256537(n) - 1), n >= 3. - _Omar E. Pol_, Apr 04 2015
%t A151723 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 *)
%t A151723 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 *)
%Y A151723 Cf. A147562, A151724, A151725, A161206, A161644, A169779, A169780, A170898, A170905, A182632, A182840, A256536, A256537.
%K A151723 nonn
%O A151723 0,3
%A A151723 _David Applegate_ and _N. J. A. Sloane_, Jun 13 2009
%E A151723 Edited by _N. J. A. Sloane_, Jan 10 2010