A266753 Decimal representation of the n-th iteration of the "Rule 163" elementary cellular automaton starting with a single ON (black) cell.

1, 4, 18, 74, 298, 1194, 4778, 19114, 76458, 305834, 1223338, 4893354, 19573418, 78293674, 313174698, 1252698794, 5010795178, 20043180714, 80172722858, 320690891434, 1282763565738, 5131054262954, 20524217051818, 82096868207274, 328387472829098

Offset: 0

Author

Robert Price, Jan 17 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. A263919, A266752.

Programs

Mathematica

rule=163; 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[FromDigits[catri[[k]],2],{k,1,rows}]   (* Decimal Representation of Rows *)

Formula

Conjectures from Colin Barker, Jan 20 2016 and Apr 20 2019: (Start)

a(n) = 5*a(n-1)-4*a(n-2) for n>2.

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

(End)

Empirical a(n) = (7*4^n - 4)/6 for n>1. - Colin Barker, Nov 25 2016 and Apr 20 2019

a(n) = 4*a(n-1) + 2, n>1, conjectured. - Yosu Yurramendi, Jan 22 2017

a(n) = 2*A020988(n) - A020988(n-1) = A020988(n) + 2^(2n-1) for n > 0, conjectured. - Yosu Yurramendi, Jan 24 2017 [n range correction - Karl V. Keller, Jr., May 07 2022]

a(n) = A072197(n-1) + A002450(n), n > 0, conjectured. - Yosu Yurramendi, Mar 03 2017

Extensions

Removed an unjustified claim that Colin Barker's conjectures are correct. Removed a program based on a conjecture. - N. J. A. Sloane, Jun 13 2022