A205975 a(n) = Fibonacci(n)*A002652(n) for n>=1, with a(0)=1, where A002652 lists the coefficients in theta series of Kleinian lattice Z[(-1+sqrt(-7))/2].

1, 2, 4, 0, 18, 0, 0, 26, 168, 68, 0, 356, 0, 0, 1508, 0, 9870, 0, 10336, 0, 0, 0, 141688, 114628, 0, 150050, 0, 0, 1906866, 2056916, 0, 0, 26139708, 0, 0, 0, 89582112, 96631268, 0, 0, 0, 0, 0, 1733977748, 8416904796, 0, 14690495224, 0, 0, 15557484098

Offset

0,2

Comments

Compare the g.f. to the Lambert series of A002652:

1 + 2*Sum_{n>=1} Kronecker(n,7)*x^n/(1-x^n).

Links

Table of n, a(n) for n=0..49.

Formula

G.f.: 1 + 2*Sum_{n>=1} Fibonacci(n)*Kronecker(n,7)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).

Example

G.f.: A(x) = 1 + 2*x + 4*x^2 + 18*x^4 + 26*x^7 + 168*x^8 + 68*x^9 + 356*x^11 +...
where A(x) = 1 + 1*2*x + 1*4*x^2 + 3*6*x^4 + 14*2*x^7 + 21*8*x^8 + 34*2*x^9 + 89*4*x^11 + 377*4*x^14 + 987*10*x^16 +...+ Fibonacci(n)*A002652(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 2*( 1*x/(1-x-x^2) + 1*x^2/(1-3*x^2+x^4) - 2*x^3/(1-4*x^3-x^6) + 3*x^4/(1-7*x^4+x^8) - 5*x^5/(1-11*x^5-x^10) - 8*x^6/(1-18*x^6+x^12) + 0*13*x^7/(1+29*x^7-x^14) +...).
The values of the symbol Kronecker(n,7) repeat [1,1,-1,1,-1,-1,0, ...].

Mathematica

terms = 50; s = 1 + 2 Sum[Fibonacci[n]*KroneckerSymbol[n, 7]*x^n/(1 - LucasL[n]*x^n + (-1)^n*x^(2*n)), {n, 1, terms}] + O[x]^terms; CoefficientList[s, x] (* Jean-François Alcover, Jul 05 2017 *)

Prog

(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 + 2*sum(m=1, n, fibonacci(m)*kronecker(m, 7)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))), n)}
for(n=0, 60, print1(a(n), ", "))

Crossrefs

Cf. A002652, A205974, A205976, A203847, A000204 (Lucas).

Cf. A209455 (Pell variant).

Sequence in context: A127511 A321956 A173315 * A231915 A009170 A009625

Adjacent sequences: A205972 A205973 A205974 * A205976 A205977 A205978

Keyword

nonn

Author

Paul D. Hanna, Feb 04 2012

Status

approved