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

0,2

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

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

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 *)

Crossrefs

Cf. A263919, A266752.

Sequence in context: A202358 A075675 A047034 * A307323 A218059 A069008

Adjacent sequences: A266750 A266751 A266752 * A266754 A266755 A266756

Keyword

nonn,easy

Author

Robert Price, Jan 17 2016

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

Status

approved