bibkey: price2015rule13oncells authors: Robert Price year: 2015 title: “OEIS A266285, ON cells of the Rule 13 elementary cellular automaton: closed-form, recurrence, generating-function and parity conjectures” doi: null url: https://oeis.org/A266285 claim: “A266285 is the number of ON (black) cells in the n-th iteration of the Rule 13 elementary cellular automaton started from a single ON cell; the entry records Colin Barker’s conjectures a(n) = ((-1)^n*(3-2n)+4n+1)/4, a(n) = 2a(n-2)-a(n-4) for n>3 and g.f. (1+x+2x^3)/((1-x)^2*(1+x)^2), and Ctibor O. Zizka’s conjectures a(2n) = n+1 and a(2n+1) = 3*n+1.” strata_touched:

• D5/S3/StatisticalMechanics/CellularAutomata/Rule13OnCellCount license: citation-only triage: anchor

OEIS A266285

A266285 (Robert Price, Dec 26 2015, offset 0, data 1, 1, 2, 4, 3, 7, 4, 10, 5, 13, 6, 16, …) is

Number of ON (black) cells in the n-th iteration of the “Rule 13” elementary cellular automaton starting with a single ON (black) cell.

Its program evolves CellularAutomaton[13, {{1}, 0}, …], keeps the central 2n + 1 cells of row n and counts the ON cells. The formula field reads

Conjectures from Colin Barker, Dec 28 2015 and Apr 14 2019: (Start) a(n) = ((-1)^n*(3-2n)+4n+1)/4. a(n) = 2a(n-2)-a(n-4) for n>3. G.f.: (1+x+2x^3) / ((1-x)^2*(1+x)^2). (End) Conjecture from Ctibor O. Zizka, Mar 11 2025: (Start) a(2n) = n + 1. a(2n + 1) = 3*n + 1.(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/A266285 (revision 28, last modified 2025-03-12; name, program and formula fields; retrieved through the OEIS JSON interface).