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Recurrence: a(0)=1, a(1) = a(2) = 2; a(2^m-1)=2 for m >= 2; a(2^m) = 3 for m >= 2; a(2^m-2) = m for m >= 3; otherwise, for m >= 5, if m=2^i+j (0 <= j < 2^i - 1), a(m) = a(j) + a(j+1).
a(n) = sum_{k >= 0, n+k odd} binomial(A000120(n+k),k); the sum may be restricted further to k <= 2*A000523(n+1) [based on Hagen von Eitzen's formula for A151552].