A267459 Total number of ON (black) cells after n iterations of the "Rule 133" elementary cellular automaton starting with a single ON (black) cell.

1, 2, 3, 4, 7, 10, 15, 20, 27, 34, 43, 52, 63, 74, 87, 100, 115, 130, 147, 164, 183, 202, 223, 244, 267, 290, 315, 340, 367, 394, 423, 452, 483, 514, 547, 580, 615, 650, 687, 724, 763, 802, 843, 884, 927, 970, 1015, 1060, 1107, 1154, 1203, 1252, 1303, 1354

Offset

0,2

Comments

Identical to A105343(n-1) for n > 1. - Guenther Schrack, Jun 01 2018

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 16 2016 and Apr 17 2019: (Start)

a(n) = (2*n^2 - 4*n + (-1)^n + 11)/4 for n > 0.

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

G.f.: (1-x^2+2*x^4) / ((1-x)^3*(1+x)).

(End)

Mathematica

rule=133; 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. A267423.

Sequence in context: A056518 A160644 A347734 * A330357 A145467 A325341

Adjacent sequences: A267456 A267457 A267458 * A267460 A267461 A267462

Keyword

nonn,easy

Author

Robert Price, Jan 15 2016

Status

approved