A267581 Decimal representation of the middle column of the "Rule 167" elementary cellular automaton starting with a single ON (black) cell.

1, 3, 6, 13, 26, 53, 107, 215, 430, 861, 1723, 3447, 6895, 13791, 27583, 55167, 110334, 220669, 441339, 882679, 1765359, 3530719, 7061439, 14122879, 28245759, 56491519, 112983039, 225966079, 451932159, 903864319, 1807728639, 3615457279, 7230914558

Offset

0,2

Comments

Assuming the conjecture that the positions of the 0-bits of the middle column ("Rule 167") are given by the sequence A000051, it follows that a possible formula could be: a(n) = 2*a(n-1) + 1 - floor((1/2)^((2^(n+1)) mod n)) with a(0)=1 and a(1)=3 (Not proved, but tested up to n = 10^4). - Andres Cicuttin, Mar 29 2016

This was proved by an autonomous AI agent, see the Lean file. - Ralf Stephan, May 28 2026

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

Google Deepmind, AlphaProof Nexus: A267581 Lean file

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

a(n) = floor(c*2^(n+1)), where c = 0.841789245... - Lorenzo Sauras Altuzarra, Jan 03 2023

Mathematica

rule=167; 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 *)

Crossrefs

Cf. A000051, A267576.

Sequence in context: A072762 A081254 A125049 * A320733 A164991 A213255

Adjacent sequences: A267578 A267579 A267580 * A267582 A267583 A267584

Keyword

nonn,easy

Author

Robert Price, Jan 17 2016

Status

approved