cp's OEIS Frontend

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.

%I A067585 #10 Aug 04 2020 13:38:42
%S A067585 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,
%T A067585 20,19,17,16,18,21,23,26,24,31,29,4,6,1,3,14,12,11,9,56,58,61,63,50,
%U A067585 48,55,53,44,46,41,43,38,36,35,33,32,34,37,39,42,40,47,45,52,54,49,51,62
%N 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.
%C A067585 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).
%C A067585 For n such that a(n) = n (fixed points) cf. A083713. Iteration of the mapping leads to one of these fixed points.
%H A067585 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GameofLife.html">Game of Life</a>
%e A067585 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.
%o A067585 (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}
%o A067585 {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)),","))}
%Y A067585 Cf. A083713.
%K A067585 nonn,base
%O A067585 0,4
%A A067585 _Joseph L. Pe_, Jan 31 2002
%E A067585 Edited and extended by _Klaus Brockhaus_, Jun 14 2003