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A266223
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Total number of OFF (white) cells after n iterations of the "Rule 7" elementary cellular automaton starting with a single ON (black) cell.
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8
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0, 1, 6, 6, 15, 15, 28, 28, 45, 45, 66, 66, 91, 91, 120, 120, 153, 153, 190, 190, 231, 231, 276, 276, 325, 325, 378, 378, 435, 435, 496, 496, 561, 561, 630, 630, 703, 703, 780, 780, 861, 861, 946, 946, 1035, 1035, 1128, 1128, 1225, 1225, 1326, 1326, 1431
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OFFSET
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0,3
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REFERENCES
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S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
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LINKS
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FORMULA
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Conjectures from Colin Barker, Dec 26 2015 and Apr 14 2019: (Start)
a(n) = 1/2*(n+1)*(n+(-1)^n+1) for n>0.
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>5.
G.f.: x*(1+5*x-2*x^2-x^3+x^4) / ((1-x)^3*(1+x)^2).
(End)
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MATHEMATICA
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rule=7; 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 *) nbc=Table[Total[catri[[k]]], {k, 1, rows}]; (* Number of Black cells in stage n *) nwc=Table[Length[catri[[k]]]-nbc[[k]], {k, 1, rows}]; (* Number of White cells in stage n *) Table[Total[Take[nwc, k]], {k, 1, rows}] (* Number of White cells through stage n *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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