cp's OEIS Frontend

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].

%I A205975 #12 Jul 05 2017 02:56:33
%S A205975 1,2,4,0,18,0,0,26,168,68,0,356,0,0,1508,0,9870,0,10336,0,0,0,141688,
%T A205975 114628,0,150050,0,0,1906866,2056916,0,0,26139708,0,0,0,89582112,
%U A205975 96631268,0,0,0,0,0,1733977748,8416904796,0,14690495224,0,0,15557484098
%N 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].
%C A205975 Compare the g.f. to the Lambert series of A002652:
%C A205975 1 + 2*Sum_{n>=1} Kronecker(n,7)*x^n/(1-x^n).
%F A205975 G.f.: 1 + 2*Sum_{n>=1} Fibonacci(n)*Kronecker(n,7)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).
%e A205975 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 +...
%e A205975 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 +...
%e A205975 The g.f. is also given by the identity:
%e A205975 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) +...).
%e A205975 The values of the symbol Kronecker(n,7) repeat [1,1,-1,1,-1,-1,0, ...].
%t A205975 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 *)
%o A205975 (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A205975 {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)}
%o A205975 for(n=0,60,print1(a(n),", "))
%Y A205975 Cf. A002652, A205974, A205976, A203847, A000204 (Lucas).
%Y A205975 Cf. A209455 (Pell variant).
%K A205975 nonn
%O A205975 0,2
%A A205975 _Paul D. Hanna_, Feb 04 2012