A266724 Number of OFF (white) cells in the n-th iteration of the "Rule 59" elementary cellular automaton starting with a single ON (black) cell.

0, 1, 3, 1, 7, 1, 11, 1, 15, 1, 19, 1, 23, 1, 27, 1, 31, 1, 35, 1, 39, 1, 43, 1, 47, 1, 51, 1, 55, 1, 59, 1, 63, 1, 67, 1, 71, 1, 75, 1, 79, 1, 83, 1, 87, 1, 91, 1, 95, 1, 99, 1, 103, 1, 107, 1, 111, 1, 115, 1, 119, 1, 123, 1, 127, 1, 131, 1, 135, 1, 139, 1

Offset: 0

Author

Robert Price, Jan 03 2016

Keywords

• nonn

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
• Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
• 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. A266716.

Programs

Mathematica

rule=59; 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[Length[catri[[k]]]-nbc[[k]],{k,1,rows}] (* Number of White cells in stage n *)

Formula

Conjectures from Colin Barker, Jan 05 2016 and Apr 18 2019: (Start)

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

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

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

(End)