cp's OEIS Frontend

A292680 Rule 6: 000, ..., 111 -> 0, 1, 1, 0, 0, 0, 0, 0.

%I A292680 #10 Feb 16 2025 08:33:51
%S A292680 0,6,12,8,24,26,16,16,48,54,52,48,32,34,32,32,96,102,108,104,104,106,
%T A292680 96,96,64,70,68,64,64,66,64,64,192,198,204,200,216,218,208,208,208,
%U A292680 214,212,208,192,194,192,192,128,134,140,136,136,138,128,128,128,134,132,128
%N A292680 Rule 6: 000, ..., 111 -> 0, 1, 1, 0, 0, 0, 0, 0.
%C A292680 The orbit of 1 under this rule is A266180.
%C A292680 Rule 6 is the smallest rule which is even (otherwise infinitely many bits would be switched on at step 1, for any finite starting value) and nontrivial (i.e., does not lead to extinction nor simple reproduction, possibly shifted left or right, of a single-bit initial state).
%C A292680 As is customary in the context of elementary cellular automata, the result is the bitmap obtained from the argument extended by one bit to the right (as to consider the cell which has bit 0 of the input as left neighbor), cf. example. Since the rule has a value < 16, no cell having its left neighbor 'on' will be on. Therefore all values a(n) are even. See A292681 for the variant without this extension beyond bit 0, i.e., a(n)/2.
%H A292680 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A292680 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A292680 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%e A292680      n        |          a(n)
%e A292680    0 =   0[2] |       0[2] =  0
%e A292680    1 =   1[2] |     110[2] =  6 (bits below 001 and 010 are on, below 100 is off)
%e A292680    2 =  10[2] |    1100[2] = 12 (as above, plus an additional bit 0 below 000)
%e A292680    3 =  11[2] |    1000[2] =  8 (1 below 001, 0 below 011, 110 and 100.)
%e A292680    4 = 100[2] |   11000[2] = 24 (as n = 1 and n = 2, shifted right once more)
%e A292680    5 = 101[2] |   11010[2] = 26 (1 below 001 and 010 (twice), 0 below 101 and 100)
%e A292680    6 = 110[2] |   10000[2] = 16 (as n = 3, shifted right once)
%e A292680    7 = 111[2] |   10000[2] = 16 (1 below 001, 0 below 011, 111, 110 and 100).
%o A292680 (PARI) apply( A292680(n,r=6)=sum(i=0,logint(!n+n<<=2,2)+1,bittest(r,(n>>i)%8)<<i),[0..30])
%Y A292680 Cf. A266178, A266179, A266180, A019590, A003953, A003945, A000034, A032766, A042948, A035608.
%K A292680 nonn,easy,base
%O A292680 0,2
%A A292680 _M. F. Hasler_, Oct 09 2017