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A266795
Total number of OFF (white) cells after n iterations of the "Rule 61" elementary cellular automaton starting with a single ON (black) cell.
2
0, 1, 4, 6, 11, 15, 23, 27, 39, 43, 59, 63, 83, 87, 111, 115, 143, 147, 179, 183, 219, 223, 263, 267, 311, 315, 363, 367, 419, 423, 479, 483, 543, 547, 611, 615, 683, 687, 759, 763, 839, 843, 923, 927, 1011, 1015, 1103, 1107, 1199, 1203, 1299, 1303, 1403
OFFSET
0,3
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
FORMULA
Conjectures from Colin Barker, Jan 04 2016 and Apr 17 2019: (Start)
a(n) = (2*n*(n+(-1)^n+1)-7*(-1)^n+3)/4 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+3*x-x^3+x^4+x^5-2*x^6+x^7) / ((1-x)^3*(1+x)^2).
(End)
MATHEMATICA
rule=61; 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 *)
CROSSREFS
Cf. A266786.
Partial sums of A266794.
Sequence in context: A084263 A232807 A309160 * A355759 A060180 A190499
KEYWORD
nonn,easy
AUTHOR
Robert Price, Jan 03 2016
STATUS
approved