cp's OEIS Frontend

A356195 The binary expansion of a(n) is obtained by applying the totalistic cellular automaton with rule 2*n to the binary expansion of n.

%I A356195 #13 Jul 31 2022 19:54:25
%S A356195 0,1,0,3,0,6,3,7,0,14,3,14,0,9,7,15,0,30,3,30,0,25,7,30,0,16,12,29,7,
%T A356195 23,15,31,0,62,3,62,0,57,7,62,0,48,12,61,7,55,15,62,0,32,28,60,7,38,
%U A356195 28,61,0,33,19,51,15,47,31,63,0,126,3,126,0,121,7,126
%N A356195 The binary expansion of a(n) is obtained by applying the totalistic cellular automaton with rule 2*n to the binary expansion of n.
%C A356195 To compute the binary expansion of a(n):
%C A356195 - we scan the binary digits of n from right to left,
%C A356195 - at some position k >= 0 (0 corresponding to the least significant bit):
%C A356195      - we count the number of 1's at positions >= k, say we have w 1's,
%C A356195      - if 2^w appears in the binary expansion of 2*n,
%C A356195        then we insert a 1,
%C A356195        otherwise we insert a 0,
%C A356195      - as we are considering an even automaton (with rule 2*n),
%C A356195        once scanning the leading 0's of n, we will only insert 0's,
%C A356195      - and the result will have finitely many 1's.
%C A356195 More formally: 2^k appears in the binary expansion of a(n) iff 2^A000120(floor(n/2^k)) appears in the binary expansion of 2*n.
%H A356195 Rémy Sigrist, <a href="/A356195/b356195.txt">Table of n, a(n) for n = 0..8192</a>
%H A356195 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TotalisticCellularAutomaton.html">Totalistic Cellular Automaton</a>
%H A356195 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A356195 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A356195 a(n) = n iff n belongs to A000225.
%F A356195 a(n) = 0 iff n AND A038573(n) = 0 (where AND denotes the bitwise AND operator).
%e A356195 For n = 43:
%e A356195 - the binary expansion of 2*43 is "1010110",
%e A356195 - so we apply the following totalistic cellular automaton:
%e A356195        w   | >=7  6   5   4   3   2   1   0
%e A356195        out |  0   1   0   1   0   1   1   0
%e A356195 - scanning the binary expansion of n, we obtains:
%e A356195     bin(n)    | 1   0   1   0   1   1
%e A356195     w         | 1   1   2   2   3   4
%e A356195     bin(a(n)) | 1   1   1   1   0   1
%e A356195 - so a(n) = 61.
%o A356195 (PARI) a(n) = { my (v=0, m=n); for (k=0, oo, if (m==0, return (v), bittest(2*n, hammingweight(m)), v+=2^k); m\=2) }
%Y A356195 Cf. A000120, A000225, A038573, A352528, A356215.
%K A356195 nonn,base
%O A356195 0,4
%A A356195 _Rémy Sigrist_, Jul 29 2022