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A360296 a(1) = 1, and for any n > 1, a(n) is the sum of the terms of the sequence at indices k < n whose binary digits appear in order but not necessarily as consecutive digits in the binary representation of n.

%I A360296 #8 Feb 02 2023 14:42:44
%S A360296 1,1,1,2,3,3,2,4,8,11,8,8,11,8,4,8,20,34,26,34,51,40,20,20,40,51,34,
%T A360296 26,34,20,8,16,48,96,76,118,186,152,76,96,208,281,186,152,208,124,48,
%U A360296 48,124,208,152,186,281,208,96,76,152,186,118,76,96,48,16,32
%N A360296 a(1) = 1, and for any n > 1, a(n) is the sum of the terms of the sequence at indices k < n whose binary digits appear in order but not necessarily as consecutive digits in the binary representation of n.
%C A360296 This sequence is a variant of A165418.
%H A360296 Rémy Sigrist, <a href="/A360296/b360296.txt">Table of n, a(n) for n = 1..8192</a>
%H A360296 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A360296 a(n) = Sum_{k = 1..A301977(n-1)} a(A301983(n, k)) for any n > 1.
%F A360296 a(2^k) = 2^(k-1) for any k > 0.
%F A360296 a(2^k-1) = 2^(k-2) for any k > 1.
%F A360296 a(n) >= A165418(n).
%e A360296 The first terms, alongside the corresponding k's, are:
%e A360296   n   a(n)  k's
%e A360296   --  ----  ------------------
%e A360296    1     1  N/A
%e A360296    2     1  {1}
%e A360296    3     1  {1}
%e A360296    4     2  {1, 2}
%e A360296    5     3  {1, 2, 3}
%e A360296    6     3  {1, 2, 3}
%e A360296    7     2  {1, 3}
%e A360296    8     4  {1, 2, 4}
%e A360296    9     8  {1, 2, 3, 4, 5}
%e A360296   10    11  {1, 2, 3, 4, 5, 6}
%e A360296   11     8  {1, 2, 3, 5, 7}
%e A360296   12     8  {1, 2, 3, 4, 6}
%e A360296   13    11  {1, 2, 3, 5, 6, 7}
%e A360296   14     8  {1, 2, 3, 6, 7}
%e A360296   15     4  {1, 3, 7}
%e A360296   16     8  {1, 2, 4, 8}
%o A360296 (PARI) { for (n=1, #a=vector(64), print1 (a[n]=if (n==1, 1, s = [1]; b = binary(n); for (k=2, #b, s = setunion(s, apply(v -> 2*v+b[k], s))); sum(k=1, #s-1, a[s[k]]);)", ")) }
%Y A360296 Cf. A165418, A301983.
%K A360296 nonn,look,base
%O A360296 1,4
%A A360296 _Rémy Sigrist_, Feb 02 2023