A266722 Number of ON (black) cells in the n-th iteration of the "Rule 59" elementary cellular automaton starting with a single ON (black) cell.
1, 2, 2, 6, 2, 10, 2, 14, 2, 18, 2, 22, 2, 26, 2, 30, 2, 34, 2, 38, 2, 42, 2, 46, 2, 50, 2, 54, 2, 58, 2, 62, 2, 66, 2, 70, 2, 74, 2, 78, 2, 82, 2, 86, 2, 90, 2, 94, 2, 98, 2, 102, 2, 106, 2, 110, 2, 114, 2, 118, 2, 122, 2, 126, 2, 130, 2, 134, 2, 138, 2
Offset: 0
Links
- Robert Price, Table of n, a(n) for n = 0..1000
- Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
- Eric Weisstein's World of Mathematics, Gear Graph
- Eric Weisstein's World of Mathematics, Matching
- Eric Weisstein's World of Mathematics, Maximum Independent Edge Set
- Eric Weisstein's World of Mathematics, Maximum Independent Vertex Set
- Eric Weisstein's World of Mathematics, Minimum Dominating Set
- Eric Weisstein's World of Mathematics, Prism Graph
- Eric Weisstein's World of Mathematics, Web Graph
- Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
- Index entries for sequences related to cellular automata
- Index to Elementary Cellular Automata
- Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).
Programs
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Mathematica
rule=59; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[Total[catri[[k]]],{k,1,rows}] (* Number of Black cells in stage n *) Join[{1}, Table[Piecewise[{{2, Mod[n, 2] == 0}, {2 n, Mod[n, 2] == 1}}], {n, 20}]] (* Eric W. Weisstein, Sep 09 2021 *) Join[{1}, LinearRecurrence[{0, 2, 0, -1}, {2, 2, 6, 2}, 20]] (* Eric W. Weisstein, Sep 09 2021 *) CoefficientList[Series[(1 + 2 x + 2 x^3 - x^4)/(-1 + x^2)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 09 2021 *)
Formula
From Colin Barker, Jan 05 2016 and Apr 17 2019: (Start)
a(n) = 1+(-1)^n+n-(-1)^n*n for n>0.
a(n) = 2*a(n-2)-a(n-4) for n>4.
G.f.: (1+2*x-x^2)*(1+x^2) / ((1-x)^2*(1+x)^2). (End)
Comments