A263807 Total number of ON (black) cells after n iterations of the "Rule 157" elementary cellular automaton starting with a single ON (black) cell.

1, 3, 6, 11, 17, 25, 34, 45, 57, 71, 86, 103, 121, 141, 162, 185, 209, 235, 262, 291, 321, 353, 386, 421, 457, 495, 534, 575, 617, 661, 706, 753, 801, 851, 902, 955, 1009, 1065, 1122, 1181, 1241, 1303, 1366, 1431, 1497, 1565, 1634, 1705, 1777, 1851, 1926

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 20 2016 and Apr 16 2019: (Start)

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>3.

G.f.: (1+x+x^3) / ((1-x)^3*(1+x)).

(End)

Mathematica

rule=157; 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 *) Table[Total[Take[nbc, k]], {k, 1, rows}] (* Number of Black cells through stage n *)

Crossrefs

Cf. A263804.

Sequence in context: A022775 A025743 A022338 * A176708 A273140 A320272

Adjacent sequences: A263804 A263805 A263806 * A263808 A263809 A263810

Keyword

nonn,easy

Author

Robert Price, Jan 17 2016

Status

approved