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%I #5 Jul 10 2019 21:25:04
%S 1,1,4,1,16,4,4,1,64,16,16,4,16,4,4,1,256,64,64,16,64,16,16,4,64,16,
%T 16,4,16,4,4,1,1024,256,256,64,256,64,64,16,256,64,64,16,64,16,16,4,
%U 256,64,64,16,64,16,16,4,64,16,16,4,16,4,4,1,4096,1024,1024,256,1024,256,256,64,1024,256,256,64
%N a(0) = 1; a(2*n) = 4*a(n), a(2*n+1) = a(n).
%F G.f. A(x) satisfies: A(x) = (4 + x) * A(x^2) - 3.
%F a(0) = 1; for n > 0, a(n) = 4^(number of 0's in binary representation of n).
%t a[0] = 1; a[n_] := If[EvenQ[n], 4 a[n/2], a[(n - 1)/2]]; Table[a[n], {n, 0, 75}]
%t nmax = 75; A[_] = 1; Do[A[x_] = (4 + x) A[x^2] - 3 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t Join[{1}, Table[4^Count[IntegerDigits[n, 2], 0], {n, 1, 75}]]
%Y Cf. A000225 (positions of 1's), A000302, A023416, A080100, A080791, A102376, A309057.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jul 10 2019
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