A266596 Total number of OFF (white) cells after n iterations of the "Rule 37" elementary cellular automaton starting with a single ON (black) cell.

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.

Links

Robert Price, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

S. Wolfram, A New Kind of Science

Index entries for sequences related to cellular automata

Index to Elementary Cellular Automata

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

Adjacent sequences: A266593 A266594 A266595 * A266597 A266598 A266599

Keyword

nonn,easy

Author

Robert Price, Jan 01 2016

Status

approved