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A265429
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Total number of ON (black) cells after n iterations of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell.
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3
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1, 3, 5, 9, 13, 18, 23, 30, 37, 45, 53, 63, 73, 84, 95, 108, 121, 135, 149, 165, 181, 198, 215, 234, 253, 273, 293, 315, 337, 360, 383, 408, 433, 459, 485, 513, 541, 570, 599, 630, 661, 693, 725, 759, 793, 828, 863, 900, 937, 975, 1013, 1053, 1093, 1134
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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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Conjectures from Colin Barker, Dec 09 2015 and Apr 16 2019: (Start)
a(n) = (1/16)*(6*n^2 + 24*n - 3*(-1)^n + 2*(-i)^n + 2*i^n + 15) where i = sqrt(-1).
G.f.: (1 + x + 2*x^3 - x^4) / ((1-x)^3*(1+x)*(1+x^2)).
(End)
Conjecture: the sequence consists of all numbers k > 0 such that floor(sqrt(8*(k+1)/3)) != floor(sqrt(8*k/3)). - Gevorg Hmayakyan, Sep 01 2019
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EXAMPLE
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First 12 rows, replacing "0" with ".", ignoring "0" outside of range of 1's for better visibility of ON cells, followed by total number of ON cells per row, and running total up to that row:
1 = 1 -> 1
1 1 = 2 -> 3
1 . 1 = 2 -> 5
1 1 1 1 = 4 -> 9
1 1 1 . 1 = 4 -> 13
1 1 . 1 1 1 = 5 -> 18
1 . 1 1 1 . 1 = 5 -> 23
1 1 1 1 . 1 1 1 = 7 -> 30
1 1 1 . 1 1 1 . 1 = 7 -> 37
1 1 . 1 1 1 . 1 1 1 = 8 -> 45
1 . 1 1 1 . 1 1 1 . 1 = 8 -> 53
1 1 1 1 . 1 1 1 . 1 1 1 = 10 -> 63
1 1 1 . 1 1 1 . 1 1 1 . 1 = 10 -> 72
(End)
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MATHEMATICA
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rule = 188; rows = 30; Table[Total[Take[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}], k]], {k, 1, rows}]
Accumulate[Count[#, n_ /; n == 1] & /@ CellularAutomaton[188, {{1}, 0}, 53]] (* Michael De Vlieger, Dec 09 2015 *)
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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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