A357366 Expansion of Product_{k>=0} 1 / (1 - x^(2^k) - x^(2^(k+1)))^(2^k).
1, 1, 4, 5, 18, 23, 59, 82, 203, 285, 610, 895, 1838, 2733, 5217, 7950, 14763, 22713, 40526, 63239, 110652, 173891, 297529, 471420, 796706, 1268126, 2116508, 3384634, 5606444, 8991078, 14791302, 23782380, 38955441, 62737821, 102388280, 165126101, 268844542, 433970643
Offset: 0
Keywords
Programs
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Mathematica
nmax = 37; CoefficientList[Series[Product[1/(1 - x^(2^k) - x^(2^(k + 1)))^(2^k), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x] nmax = 37; A[] = 1; Do[A[x] = A[x^2]^2/(1 - x - x^2) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Formula
G.f. A(x) satisfies: A(x) = A(x^2)^2 / (1 - x - x^2).
a(n) ~ c * phi^(n+1) / sqrt(5), where c = Product_{k>=1} 1/(1 - x^(2^k) - x^(2^(k+1)))^(2^k) = 11.1991985012843182084779984477952870732899201240395056... and phi = A001622 is the golden ratio. - Vaclav Kotesovec, Oct 08 2022