A268656 G.f.: Sum_{n=-oo..+oo} ( x^n / (1 - x^(2*n)*(1+x)^(n+1)) )^2.

1, 0, 0, 0, 2, 0, 6, 6, 11, 16, 38, 58, 103, 152, 267, 486, 861, 1392, 2276, 3812, 6562, 11224, 18906, 31446, 52392, 87776, 147423, 246698, 411358, 685988, 1145964, 1913208, 3184968, 5288788, 8779115, 14587396, 24250424, 40280516, 66808605, 110689870, 183321646, 303558816, 502460050, 831201108, 1374306023, 2271447536, 3753011555, 6198348494, 10232046899, 16883378468, 27848975062, 45923355280

Offset

-2,5

Links

Paul D. Hanna, Table of n, a(n) for n = -2..400

Formula

G.f.: [ Sum_{n=-oo..+oo} x^n / (1 - x^(2*n)*(1+x)^(n+1)) ]^2.

a(n) ~ ((1+sqrt(5))/2)^n * n / 20. - Vaclav Kotesovec, Mar 10 2016

Example

G.f.: A(x) = 1/x^2 + 2*x^2 + 6*x^4 + 6*x^5 + 11*x^6 + 16*x^7 + 38*x^8 + 58*x^9 + 103*x^10 + 152*x^11 + 267*x^12 + 486*x^13 + 861*x^14 + 1392*x^15 + 2276*x^16 +...
such that the g.f. equals the sum of the squares
A(x) = Sum_{n=-oo..+oo} [ x^n / (1 - x^(2*n)*(1+x)^(n+1)) ]^2
and also the g.f. equals the square of the sum
A(x) = [ Sum_{n=-oo..+oo} x^n / (1 - x^(2*n)*(1+x)^(n+1)) ]^2
Thus
A(x)^(1/2) = Sum_{n=-oo..+oo} x^n / (1 - x^(2*n)*(1+x)^(n+1))
explicitly,
A(x)^(1/2) = 1/x + x^3 + 3*x^5 + 3*x^6 + 5*x^7 + 8*x^8 + 16*x^9 + 26*x^10 + 42*x^11 + 59*x^12 + 98*x^13 + 178*x^14 + 304*x^15 + 471*x^16 + 724*x^17 + 1167*x^18 + 1960*x^19 + 3266*x^20 + 5311*x^21 + 8521*x^22 + 13723*x^23 + 22288*x^24 + 36283*x^25 + 58825*x^26 + 95187*x^27 + 154542*x^28 + 251751*x^29 + 409437*x^30 +...

Prog

(PARI) {a(n) = my(A); A = sum(m=-2*n-4, n+2, ( x^m / (1 - x^(2*m)*(1+x)^(m+1)) +x^3*O(x^n) )^2 )^1; polcoeff(x^2*A, n+2)}
for(n=-2, 60, print1(a(n), ", "))
(PARI) {a(n) = my(A); A = sum(m=-2*n-4, n+2, ( x^m / (1 - x^(2*m)*(1+x)^(m+1)) +x^3*O(x^n) )^1 )^2; polcoeff(x^2*A, n+2)}
for(n=-2, 60, print1(a(n), ", "))

Crossrefs

Sequence in context: A106848 A059444 A328473 * A318619 A340484 A341930

Adjacent sequences: A268653 A268654 A268655 * A268657 A268658 A268659

Keyword

nonn

Author

Paul D. Hanna, Mar 09 2016

Status

approved