A267045 Decimal representation of the middle column of the "Rule 91" elementary cellular automaton starting with a single ON (black) cell.

1, 2, 4, 8, 17, 34, 69, 138, 277, 554, 1109, 2218, 4437, 8874, 17749, 35498, 70997, 141994, 283989, 567978, 1135957, 2271914, 4543829, 9087658, 18175317, 36350634, 72701269, 145402538, 290805077, 581610154, 1163220309, 2326440618, 4652881237, 9305762474

Offset: 0

Author

Robert Price, Jan 09 2016

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..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. A267015, A267044.

Programs

Mathematica

rule=91; 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 *) mc=Table[catri[[k]][[k]],{k,1,rows}]; (* Keep only middle cell from each row *) Table[FromDigits[Take[mc,k],2],{k,1,rows}]  (* Binary Representation of Middle Column *)

Formula

Conjectures from Colin Barker, Jan 10 2016 and Apr 17 2019: (Start)

a(n) = (2*(-1)^n+13*2^n-6)/12 for n>1.

a(n) = 2*a(n-1)+a(n-2)-2*a(n-3) for n>4. [Typo corrected by Karl V. Keller, Jr., Mar 16 2022]

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

(End)