cp's OEIS Frontend

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.

A290114 Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood.

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%I A290114 #35 Feb 16 2025 08:33:49
%S A290114 1,3,5,11,23,47,95,191,383,767,1535,3071,6143,12287,24575,49151,98303,
%T A290114 196607,393215,786431,1572863,3145727,6291455,12582911,25165823,
%U A290114 50331647,100663295,201326591,402653183,805306367,1610612735,3221225471,6442450943,12884901887
%N A290114 Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood.
%C A290114 Initialized with a single black (ON) cell at stage zero.
%D A290114 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A290114 Robert Price, <a href="/A290114/b290114.txt">Table of n, a(n) for n = 0..126</a>
%H A290114 Robert Price, <a href="/A290114/a290114.tmp.txt">Diagrams of first 20 stages</a>
%H A290114 Robert Price, <a href="/A290114/a290114.tmp.txt">Diagrams of first 20 stages</a>
%H A290114 N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015.
%H A290114 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A290114 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A290114 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A290114 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).
%H A290114 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A290114 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A290114 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A290114 For n>1, a(n) = 3*2^(n-1)-1.
%F A290114 a(n) = A266550(n+2) for n > 1. - _Georg Fischer_, Oct 30 2018
%F A290114 a(n) = 2*a(n-1) + 1 for n=1 and n>=3. - _Gennady Eremin_, Aug 26 2023
%F A290114 From _Chai Wah Wu_, Apr 02 2024: (Start)
%F A290114 a(n) = 3*a(n-1) - 2*a(n-2) for n > 3.
%F A290114 G.f.: (2*x^3 - 2*x^2 + 1)/((x - 1)*(2*x - 1)). (End)
%t A290114 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A290114 code = 643; stages = 128;
%t A290114 rule = IntegerDigits[code, 2, 10];
%t A290114 g = 2 * stages + 1; (* Maximum size of grid *)
%t A290114 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A290114 ca = a;
%t A290114 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A290114 PrependTo[ca, a];
%t A290114 (* Trim full grid to reflect growth by one cell at each stage *)
%t A290114 k = (Length[ca[[1]]] + 1)/2;
%t A290114 ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
%t A290114 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
%Y A290114 Essentially the same as A153893, A083329, A055010, A052940, A266550.
%Y A290114 Cf. A290111, A290112, A290113, .
%K A290114 nonn,easy
%O A290114 0,2
%A A290114 _Robert Price_, Jul 19 2017