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

0,4

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.

Links

Table of n, a(n) for n=0..76.

Eric Weisstein's World of Mathematics, Game of Life

Example

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.

Prog

(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)), ", "))}

Crossrefs

Cf. A083713.

Sequence in context: A319830 A309680 A010604 * A173787 A116191 A257303

Adjacent sequences: A067582 A067583 A067584 * A067586 A067587 A067588

Keyword

nonn,base

Author

Joseph L. Pe, Jan 31 2002

Extensions

Edited and extended by Klaus Brockhaus, Jun 14 2003

Status

approved