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A265722
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Number of ON (black) cells in the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.
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2
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1, 0, 1, 4, 1, 8, 1, 12, 1, 16, 1, 20, 1, 24, 1, 28, 1, 32, 1, 36, 1, 40, 1, 44, 1, 48, 1, 52, 1, 56, 1, 60, 1, 64, 1, 68, 1, 72, 1, 76, 1, 80, 1, 84, 1, 88, 1, 92, 1, 96, 1, 100, 1, 104, 1, 108, 1, 112, 1, 116, 1, 120, 1, 124, 1, 128, 1, 132, 1, 136, 1, 140
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OFFSET
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0,4
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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 14 2015 and Apr 16 2019: (Start)
a(n) = 1/2*(-2*(-1)^n*n+2*n+3*(-1)^n-1).
a(n) = 2*a(n-2) - a(n-4) for n>3.
G.f.: (1-x^2+4*x^3) / ((1-x)^2*(1+x)^2).
(End)
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EXAMPLE
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First 12 rows, replacing zeros with "." for better visibility of ON cells, followed by the total number of 1's per row at right:
1 = 1
. . . = 0
. . 1 . . = 1
1 1 . . . 1 1 = 4
. . . . 1 . . . . = 1
1 1 1 1 . . . 1 1 1 1 = 8
. . . . . . 1 . . . . . . = 1
1 1 1 1 1 1 . . . 1 1 1 1 1 1 = 12
. . . . . . . . 1 . . . . . . . . = 1
1 1 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 1 = 16
. . . . . . . . . . 1 . . . . . . . . . . = 1
1 1 1 1 1 1 1 1 1 1 . . . 1 1 1 1 1 1 1 1 1 1 = 20
(End)
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MATHEMATICA
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rule = 1; rows = 30; Table[Total[Table[Take[CellularAutomaton[rule, {{1}, 0}, rows-1, {All, All}][[k]], {rows-k+1, rows+k-1}], {k, 1, rows}][[k]]], {k, 1, rows}]
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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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STATUS
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approved
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