bibkey: price2016rule41oncells authors: Robert Price year: 2016 title: “OEIS A266614, ON cells of the Rule 41 elementary cellular automaton: recurrence and generating-function conjectures” doi: null url: https://oeis.org/A266614 claim: “A266614 is the number of ON (black) cells in the n-th iteration of the Rule 41 elementary cellular automaton started from a single ON cell; the entry records Colin Barker’s conjectures a(n) = a(n-2)+a(n-4)-a(n-6) for n>5 and g.f. (1+x^2+3x^3-2x^4+5x^5)/((1-x)^2(1+x)^2*(1+x^2)).” strata_touched:

• D5/S3/StatisticalMechanics/CellularAutomata/Rule41OnCellCount license: citation-only triage: anchor

OEIS A266614

A266614 (Robert Price, Jan 01 2016, offset 0, data 1, 0, 2, 3, 1, 8, 2, 11, 1, 16, 2, 19, …) is

Number of ON (black) cells in the n-th iteration of the “Rule 41” elementary cellular automaton starting with a single ON (black) cell.

Its program evolves CellularAutomaton[41, {{1}, 0}, …], keeps the central 2n + 1 cells of row n and counts the ON cells. The formula field reads

Conjectures from Colin Barker, Jan 02 2016 and Apr 18 2019: (Start) a(n) = a(n-2)+a(n-4)-a(n-6) for n>5. G.f.: (1+x^2+3x^3-2x^4+5x^5) / ((1-x)^2(1+x)^2*(1+x^2)). (End)

The references are S. Wolfram, A New Kind of Science, Wolfram Media, 2002, p. 55, and the MathWorld page on elementary cellular automata.

Verified locator

• URL: https://oeis.org/A266614 (revision 18, last modified 2025-02-16; name, program and formula fields; retrieved through the OEIS JSON interface).