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A267461
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Total number of OFF (white) cells after n iterations of the "Rule 133" elementary cellular automaton starting with a single ON (black) cell.
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1
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0, 2, 6, 12, 18, 26, 34, 44, 54, 66, 78, 92, 106, 122, 138, 156, 174, 194, 214, 236, 258, 282, 306, 332, 358, 386, 414, 444, 474, 506, 538, 572, 606, 642, 678, 716, 754, 794, 834, 876, 918, 962, 1006, 1052, 1098, 1146, 1194, 1244, 1294, 1346, 1398, 1452
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OFFSET
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0,2
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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) = (2*n^2+12*n-(-1)^n-7)/4 for n>0.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>4.
G.f.: 2*x*(1+x-x^3) / ((1-x)^3*(1+x)).
(End)
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MATHEMATICA
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rule=133; 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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