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 A151725 #42 May 10 2025 15:19:00
%S A151725 0,1,9,13,33,37,57,77,121,125,145,165,209,237,297,373,465,469,489,509,
%T A151725 553,581,641,717,809,837,897,981,1097,1213,1409,1645,1833,1837,1857,
%U A151725 1877,1921,1949,2009,2085,2177,2205,2265,2349,2465,2581,2777,3013
%N A151725 Number of ON states after n generations of cellular automaton rule described by the rulestring B1/S012345678.
%C A151725 A cell is turned ON if exactly one of its eight neighbors is ON. An ON cell remains ON forever.
%C A151725 We start with a single ON cell.
%C A151725 Analog of A147562, which is the case when each cell has only four neighbors.
%C A151725 The equivalent Mathematica cellular automaton is obtained with neighborhood weights {{1,1,1},{1,9,1},{1,1,1}}, rule number 261634, and starting configuration {{1}}. [_John W. Layman_, Sep 11 2009]
%C A151725 Observation: Visual pattern similar to the toothpick structure (see A139250). [_Omar E. Pol_, Dec 14 2009]
%H A151725 David Applegate, <a href="/A151725/b151725.txt">Table of n, a(n) for n = 0..1000</a>
%H A151725 David Applegate, <a href="/A139250/a139250.anim.html">The movie version</a>
%H A151725 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 A151725 N. H. Packard and S. Wolfram, <a href="http://doi.org/10.1007/BF01010423">Two-Dimensional Cellular Automata</a>, Journal of Statistical Physics, 38 (1985), 901-946. (See page 920, Figure 7e). <a href="http://new.math.uiuc.edu/im2008/dakkak/papers/files/wolfram.2dca.pdf">Alternative copy</a>
%H A151725 Rémy Sigrist, <a href="/A151725/a151725.png">Illustration of the structure at stage 513</a>
%H A151725 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>
%F A151725 For a recurrence see the Applegate-Pol-Sloane paper.
%t A151725 RasterGraphics[state_?MatrixQ, colors_Integer : 2, opts___] := Graphics[Raster[ Reverse[1 - state/(colors - 1)]], AspectRatio -> (AspectRatio /. {opts} /. AspectRatio -> Automatic), Frame -> True, FrameTicks -> None, GridLines -> None]; wt = {{1,1,1}, {1,9,1}, {1,1,1}}; rule= 261634; init={{1}}; Show[GraphicsArray[Map[RasterGraphics, CellularAutomaton[{rule, {2, wt}, {1, 1}}, {init, 0}, 9, -10]]]];nx = 100; ca = CellularAutomaton[{rule, {2, wt}, {1, 1}}, {init, 0}, nx - 1, -nx]; a = Table[Total[ca[[i]], 2], {i, 1, nx}] (* _John W. Layman_, Sep 11 2009 *)
%t A151725 A151725[0] = 0; A151725[n_] := Total[CellularAutomaton[{174766, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}, {{{1}}, 0}, {{{n - 1}}}], 2]; Array[A151725, 48, 0] (* _JungHwan Min_, Sep 01 2016 *)
%t A151725 A151725L[n_] := Prepend[Total[#, 2] & /@ CellularAutomaton[{174766, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}, {{{1}}, 0}, n - 1], 0]; A151725L[47] (* _JungHwan Min_, Sep 01 2016 *)
%Y A151725 Cf. A139250, A151726, A147562, A147582, A151723, A151735, A151747, A151728.
%Y A151725 See A151731, A151732, A151733, A151734 for the same CA except that two neighbors must be ON for a cell to turn ON.
%K A151725 nonn
%O A151725 0,3
%A A151725 _David Applegate_ and _N. J. A. Sloane_, Jun 13 2009
%E A151725 Definition clarified by _SiYang Hu_, May 10 2025