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bibkey: price2016rule201rows authors: Robert Price year: 2016 title: “OEIS A267681 and A267680, rows of the Rule 201 elementary cellular automaton: recurrence, generating-function and closed-form conjectures” doi: null url: https://oeis.org/A267681 claim: “A267681 is the decimal representation of the n-th iteration of the Rule 201 elementary cellular automaton started from a single ON cell, and A267680 its binary representation; the entries record Colin Barker’s conjectures a(n) = 5a(n-1)-20a(n-3)+16a(n-4) for n>4 with g.f. (1-5x+21x^2+14x^3-40x^4)/((1-x)(1-2x)(1+2x)(1-4x)) for A267681, a(n) = 101a(n-1)-10100a(n-3)+10000a(n-4) for n>4 with g.f. (1-101x+10101x^2+89910x^3-101000x^4)/((1-x)(1-10x)(1+10x)(1-100x)) for A267680, and M. F. Hasler’s conjecture a(n) = 24^n - (n%22 + [n]*5)*2^(n-1) - 1 for A267681.” strata_touched:

  • D5/S3/StatisticalMechanics/CellularAutomata/Rule201Rows license: citation-only triage: anchor

OEIS A267681 and A267680

A267681 (Robert Price, Jan 19 2016, offset 0, data 1, 0, 21, 99, 471, 1935, …) is

Decimal representation of the n-th iteration of the “Rule 201” elementary cellular automaton starting with a single ON (black) cell.

Its program takes the central 2n + 1 cells of row n of CellularAutomaton[201, {{1}, 0}, …] and applies FromDigits in base 2. The formula field reads

Conjectures from Colin Barker, Jan 19 2016: (Start) a(n) = 5a(n-1)-20a(n-3)+16a(n-4) for n>4. G.f.: (1-5x+21x^2+14x^3-40x^4) / ((1-x)(1-2x)(1+2x)(1-4x)). (End) Conjecture: a(n) = 24^n - (n%2*2 + [n]*5)*2^(n-1) - 1, where [n] = 1 iff n > 0; n%2 = 1 iff n is odd. - M. F. Hasler, Jul 28 2018

and the history records

Removed an unjustified claim that Colin Barker’s conjectures are correct. Removed a program based on a conjecture. - Michael De Vlieger, Jun 13 2022

A267680 (Robert Price, Jan 19 2016) is the binary representation of the same rows: the same digits read as a decimal numeral. Its formula field reads

Conjectures from Colin Barker, Jan 19 2016 and Apr 20 2019: (Start) a(n) = 101a(n-1)-10100a(n-3)+10000a(n-4) for n>4. G.f.: (1-101x+10101x^2+89910x^3-101000x^4) / ((1-x)(1-10x)(1+10x)(1-100*x)). (End)

Rule 201 is the local update rule of the Floquet-PXP cellular automaton of Wilkinson, Klobas, Prosen and Garrahan, Phys. Rev. E 102 (2020) 062107, which applies it to alternating sublattices; the entries use the synchronous elementary automaton.

Verified locator

  • URL: https://oeis.org/A267681 (revision 30, last modified 2025-02-16; name, program, formula and history fields; retrieved 2026-09-28 through the OEIS JSON interface).
  • URL: https://oeis.org/A267680 (revision 21, last modified 2025-02-16; name, program and formula fields; retrieved 2026-09-28 through the OEIS JSON interface).