A265284 Total number of ON (black) cells after n iterations of the "Rule 94" elementary cellular automaton starting with a single ON (black) cell.

1, 4, 8, 14, 20, 28, 36, 46, 56, 68, 80, 94, 108, 124, 140, 158, 176, 196, 216, 238, 260, 284, 308, 334, 360, 388, 416, 446, 476, 508, 540, 574, 608, 644, 680, 718, 756, 796, 836, 878, 920, 964, 1008, 1054, 1100, 1148, 1196, 1246, 1296, 1348, 1400, 1454

Offset: 0

Author

Robert Price, Dec 06 2015

Keywords

• nonn
• easy

References

• S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

• Robert Price, Table of n, a(n) for n = 0..999
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• Index entries for sequences related to cellular automata
• Index to Elementary Cellular Automata

Crossrefs

Cf. A118102.

Programs

Mathematica

rule = 94; rows = 30; Table[Total[Take[Table[Total[Table[Take[CellularAutomaton[rule,{{1},0},rows-1,{All,All}][[k]],{rows-k+1,rows+k-1}],{k,1,rows}][[k]]],{k,1,rows}],k]],{k,1,rows}]

Formula

Conjectures from Colin Barker, Dec 07 2015 and Apr 16 2019: (Start)

a(n) = (2*n^2+12*n-(-1)^n+1)/4 for n>0.

a(n) = (n^2+6*n)/2 for n>1 and even.

a(n) = (n^2+6*n+1)/2 for n odd.

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

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

(End)