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A379818 a(2n+1) = a(n) for n >= 0, a(2n) = a(n) + a(n - 2^f(n)) + a(2n - 2^f(n)) + a(A025480(n-1)) for n > 0 with a(0) = 1 where f(n) = A007814(n).

%I A379818 #4 Jan 14 2025 16:11:53
%S A379818 1,1,4,1,10,4,10,1,22,10,28,4,49,10,22,1,46,22,64,10,118,28,64,4,190,
%T A379818 49,118,10,190,22,46,1,94,46,136,22,256,64,148,10,424,118,292,28,478,
%U A379818 64,136,4,661,190,478,49,796,118,256,10,1177,190,424,22,661,46
%N A379818 a(2n+1) = a(n) for n >= 0, a(2n) = a(n) + a(n - 2^f(n)) + a(2n - 2^f(n)) + a(A025480(n-1)) for n > 0 with a(0) = 1 where f(n) = A007814(n).
%F A379818 Conjecture: a(2^m*(2k+1)) = Sum_{j=0..m} (binomial(m+2, j+1) - binomial(m, j))*a(2^j*k) for m >= 0, k >= 0 with a(0) = 1.
%o A379818 (PARI) upto(n) = my(A, v1); v1 = vector(n+1, i, 0); v1[1] = 1; for(i=1, n, v1[i+1] = v1[i\2+1] + if(i%2, 0, A = 1 << valuation(i/2, 2); v1[i/2-A+1] + v1[i-A+1] + v1[i\(4*A)+1])); v1
%Y A379818 Cf. A007814, A025480, A329369, A341392, A347205.
%K A379818 nonn,base
%O A379818 0,3
%A A379818 _Mikhail Kurkov_, Jan 03 2025