A067585 Binary representation of a(n) is obtained thus: replace every digit in the binary representation of n with "1" if the sum of its neighbors is 1 and with "0" otherwise.

0, 0, 1, 3, 2, 0, 7, 5, 4, 6, 1, 3, 14, 12, 11, 9, 8, 10, 13, 15, 2, 0, 7, 5, 28, 30, 25, 27, 22, 20, 19, 17, 16, 18, 21, 23, 26, 24, 31, 29, 4, 6, 1, 3, 14, 12, 11, 9, 56, 58, 61, 63, 50, 48, 55, 53, 44, 46, 41, 43, 38, 36, 35, 33, 32, 34, 37, 39, 42, 40, 47, 45, 52, 54, 49, 51, 62

Offset: 0

Author

Joseph L. Pe, Jan 31 2002

Keywords

• nonn
• base

Comments

The result of one application of the following "game of life" rule to the binary representation of n: ("1" denotes a living cell, "0" a dead cell) A living cell survives, or a dead cell becomes alive, in the next generation iff the sum of its neighbors is 1 (sum = 0 or 2 implies death from isolation or overcrowding, respectively).

For n such that a(n) = n (fixed points) cf. A083713. Iteration of the mapping leads to one of these fixed points.

Examples

			6 (decimal) = 110 -> 111, hence a(6) = 7; 21 (decimal) = 10101 -> 00000, hence a(21) = 0. Iteration on 13 gives 13 -> 12 -> 14 -> 11 -> 3, or 1101 -> 1100 -> 1110 -> 1011 -> 11 in binary.
		

Links

• Eric Weisstein's World of Mathematics, Game of Life

Crossrefs

Cf. A083713.

Programs

PARI

{b2to10(n)=local(f,d,k); f=1; k=0; while(n>0,d=divrem(n,10); n=d[1]; k=k+f*d[2]; f=2*f); k}
{for(n=0,77,v=concat(0,binary(2*n)); s="0"; for(j=1,length(v)-2,s=concat(s,v[j]!=v[j+2])); print1(b2to10(eval(s)),","))}

Extensions

Edited and extended by Klaus Brockhaus, Jun 14 2003