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A266667
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Binary representation of the n-th iteration of the "Rule 51" elementary cellular automaton starting with a single ON (black) cell.
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2
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1, 101, 100, 1110111, 10000, 11111011111, 1000000, 111111101111111, 100000000, 1111111110111111111, 10000000000, 11111111111011111111111, 1000000000000, 111111111111101111111111111, 100000000000000, 1111111111111110111111111111111, 10000000000000000
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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, Jan 03 2016 and Apr 18 2019: (Start)
a(n) = (-1+(-1)^n+9*(-5)^n*2^(1+n)+10^(1+2*n)-(-1)^n*10^(1+2*n))/18.
G.f.: (1+111*x-8891*x^2+1000*x^3+20000*x^4) / ((1-x)*(1+x)*(1+10*x)*(1-100*x)*(1+100*x)).
(End)
Conjecture: a(n) = (10*100^n - 9*10^n - 1)/9 for odd n; a(n) = 10^n for even n. - Karl V. Keller, Jr., Oct 12 2021
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MATHEMATICA
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rule=51; 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 *) Table[FromDigits[catri[[k]]], {k, 1, rows}] (* Binary Representation of Rows *)
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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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