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

Author

Robert Price, Jan 15 2016

Keywords

• nonn
• easy

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

Crossrefs

Cf. A267423.

Programs

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 *)

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)