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A267262
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Total number of OFF (white) cells after n iterations of the "Rule 111" elementary cellular automaton starting with a single ON (black) cell.
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1
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0, 1, 3, 5, 9, 13, 20, 24, 35, 39, 54, 58, 77, 81, 104, 108, 135, 139, 170, 174, 209, 213, 252, 256, 299, 303, 350, 354, 405, 409, 464, 468, 527, 531, 594, 598, 665, 669, 740, 744, 819, 823, 902, 906, 989, 993, 1080, 1084, 1175, 1179, 1274, 1278, 1377, 1381
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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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a(n) = (n^2+(-1)^n*(n-4)+2)/2 for n>3.
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>8.
G.f.: x*(1+2*x+x^4+x^5-2*x^6+x^7) / ((1-x)^3*(1+x)^2).
(End)
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MATHEMATICA
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rule=111; 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
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AUTHOR
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STATUS
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approved
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