A266661 Decimal representation of the n-th iteration of the "Rule 47" elementary cellular automaton starting with a single ON (black) cell.

1, 6, 3, 124, 3, 2044, 3, 32764, 3, 524284, 3, 8388604, 3, 134217724, 3, 2147483644, 3, 34359738364, 3, 549755813884, 3, 8796093022204, 3, 140737488355324, 3, 2251799813685244, 3, 36028797018963964, 3, 576460752303423484, 3, 9223372036854775804, 3

Offset: 0

Author

Robert Price, Jan 02 2016

Keywords

• nonn
• easy

Links

• Robert Price, Table of n, a(n) for n = 0..1000
• 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

Crossrefs

Cf. A266255, A266659, A266660.

Programs

Mathematica

rule=47; 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 03 2016 and Apr 18 2019: (Start)

a(n) = (7*(-1)^n+2^(2*n+1)-(-1)^n*2^(2*n+1)-1)/2 for n>1.

a(n) = 17*a(n-2)-16*a(n-4) for n>5.

G.f.: (1+6*x-14*x^2+22*x^3-32*x^4+32*x^5) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)).

(End)

a(n) = A266255(n), n>1. - R. J. Mathar, Jan 10 2016

Conjecture: a(n) = 2*4^n - 4 for odd n > 1; a(n) = 3 for even n > 1. - Karl V. Keller, Jr., Oct 10 2021