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A266596
Total number of OFF (white) cells after n iterations of the "Rule 37" elementary cellular automaton starting with a single ON (black) cell.
1
0, 2, 4, 9, 15, 20, 30, 35, 49, 54, 72, 77, 99, 104, 130, 135, 165, 170, 204, 209, 247, 252, 294, 299, 345, 350, 400, 405, 459, 464, 522, 527, 589, 594, 660, 665, 735, 740, 814, 819, 897, 902, 984, 989, 1075, 1080, 1170, 1175, 1269, 1274, 1372, 1377, 1479
OFFSET
0,2
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
FORMULA
Conjectures from Colin Barker, Jan 02 2016 and Apr 18 2019: (Start)
a(n) = (n^2+4*n+(-1)^n*(n-3)-3)/2 for n>0.
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>5.
G.f.: x*(2+2*x+x^2+2*x^3-3*x^4) / ((1-x)^3*(1+x)^2).
(End)
MATHEMATICA
rule=37; 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. A266588.
Sequence in context: A343592 A036277 A042960 * A045975 A058296 A347473
KEYWORD
nonn,easy
AUTHOR
Robert Price, Jan 01 2016
STATUS
approved