A267890 Decimal representation of the n-th iteration of the "Rule 239" elementary cellular automaton starting with a single ON (black) cell.

1, 6, 31, 127, 511, 2047, 8191, 32767, 131071, 524287, 2097151, 8388607, 33554431, 134217727, 536870911, 2147483647, 8589934591, 34359738367, 137438953471, 549755813887, 2199023255551, 8796093022207, 35184372088831, 140737488355327, 562949953421311

Offset: 0

Author

Robert Price, Jan 21 2016

Keywords

• nonn
• easy

Comments

Seems to differ from A267888 and A083420 only at a(1). - R. J. Mathar, Jun 24 2025

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

Crossrefs

Cf. A267871.

Programs

Mathematica

rule=239; 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 *)

Formula

Conjectures from Colin Barker, Jan 22 2016 and Apr 17 2019: (Start)

a(n) = 5*a(n-1)-4*a(n-2) for n>3.

G.f.: (1+x+5*x^2-4*x^3) / ((1-x)*(1-4*x)).

(End)

Empirical a(n) = 2^(2*n+1) - 1 for n>1. - Colin Barker, Nov 25 2016