A266256 Number of ON (black) cells in the n-th iteration of the "Rule 11" elementary cellular automaton starting with a single ON (black) cell.

1, 1, 2, 5, 2, 9, 2, 13, 2, 17, 2, 21, 2, 25, 2, 29, 2, 33, 2, 37, 2, 41, 2, 45, 2, 49, 2, 53, 2, 57, 2, 61, 2, 65, 2, 69, 2, 73, 2, 77, 2, 81, 2, 85, 2, 89, 2, 93, 2, 97, 2, 101, 2, 105, 2, 109, 2, 113, 2, 117, 2, 121, 2, 125, 2, 129, 2, 133, 2, 137, 2, 141

Offset: 0

Author

Robert Price, Dec 25 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. A266253.

Programs

Mathematica

rule=11; 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 *) Table[Total[catri[[k]]],{k,1,rows}] (* Number of Black cells in stage n *)

Formula

Conjectures from Colin Barker, Dec 27 2015 and Apr 14 2019: (Start)

a(n) = (-2*((-1)^n-1)*n+3*(-1)^n+1)/2 for n>0.

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

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

(End)

Extensions

Conjectures from Colin Barker, Apr 14 2019