A266436 Decimal representation of the n-th iteration of the "Rule 23" elementary cellular automaton starting with a single ON (black) cell.

1, 7, 0, 127, 0, 2047, 0, 32767, 0, 524287, 0, 8388607, 0, 134217727, 0, 2147483647, 0, 34359738367, 0, 549755813887, 0, 8796093022207, 0, 140737488355327, 0, 2251799813685247, 0, 36028797018963967, 0, 576460752303423487, 0, 9223372036854775807, 0

Offset: 0

Author

Robert Price, Dec 29 2015

Keywords

• nonn
• easy

Comments

With the exception of a(1) the same as A266380, A266324 and A266218. - R. J. Mathar, Jan 10 2016

Links

• Robert Price, Table of n, a(n) for n = 0..500
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
• Index entries for sequences related to cellular automata
• Index to Elementary Cellular Automata
• Index entries for linear recurrences with constant coefficients, signature (0,17,0,-16).

Crossrefs

Cf. A241955, A266434, A266435 (binary).

Programs

Mathematica

rule=23; 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 *)

Python

print([(2*4**n-1)*(n%2) + 0**n for n in range(33)]) # Karl V. Keller, Jr., Jul 06 2021

Formula

From Colin Barker, Dec 30 2015 and Apr 15 2019: (Start)

a(n) = ((-1)^n+2^(2*n+1)-(-1)^n*2^(2*n+1)-1)/2 for n>0.

a(n) = 17*a(n-2)-16*a(n-4) for n>4.

G.f.: (1+7*x-17*x^2+8*x^3+16*x^4) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)).

(End)

a(n) = (2*4^n-1)*(n mod 2) + 0^n. - Karl V. Keller, Jr., Jul 06 2021