A205967 - OEIS

%I #17 Jul 18 2018 03:02:05

%S 1,12,36,24,252,360,288,1248,3780,408,11880,12816,12096,39144,108576,

%T 43920,367164,344952,93024,1003440,3409560,1050816,7651152,8253216,

%U 8346240,27909300,61182072,2357016,213568992,185122440,179720640,516967296,1646801604,507539232

%N a(n) = Fibonacci(n)*A008653(n) for n>=1, with a(0)=1, where A008653 is the theta series of direct sum of 2 copies of hexagonal lattice.

%C Compare g.f. to the Lambert series of A008653: 1 + 12*Sum_{n>=1} Chi(n,3)*n*x^n/(1-x^n).

%C Here Chi(n,3) = principal Dirichlet character of n modulo 3.

%H G. C. Greubel, <a href="/A205967/b205967.txt">Table of n, a(n) for n = 0..2500</a>

%F G.f.: 1 + 12*Sum_{n>=1} Fibonacci(n)*Chi(n,3)*n*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).

%e G.f.: A(x) = 1 + 12*x + 36*x^2 + 24*x^3 + 252*x^4 + 360*x^5 + 288*x^6 +...

%e where A(x) = 1 + 1*12*x + 1*36*x^2 + 2*12*x^3 + 3*84*x^4 + 5*72*x^5 + 8*36*x^6 +...+ Fibonacci(n)*A008653(n)*x^n +...

%e The g.f. is also given by the identity:

%e A(x) = 1 + 12*( 1*1*x/(1-x-x^2) + 1*2*x^2/(1-3*x^2+x^4) + 3*4*x^4/(1-7*x^4+x^8) + 5*5*x^5/(1-11*x^5-x^10) + 13*7*x^7/(1-29*x^7-x^14) + 21*8*x^8/(1-47*x^8-x^16)  +...).

%e The values of the Dirichlet character Chi(n,3) repeat [1,1,0, ...].

%t terms = 34; s = 1 + 12*Sum[Fibonacci[n]*KroneckerSymbol[n, 3]^2*n*(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 *)

%t b[n_] := If[n < 1, Boole[n == 0], 12 Sum[If[Mod[d, 3] > 0, d, 0], {d, Divisors@n}]]; Table[If[n == 0, 1, b[n]*Fibonacci[n]], {n, 0, 50}] (* _G. C. Greubel_, Jul 17 2018 *)

%o (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

%o {a(n)=polcoeff(1 + 12*sum(m=1,n,fibonacci(m)*kronecker(m,3)^2*m*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}

%o for(n=0,50,print1(a(n),", "))

%Y Cf. A008653, A205966, A205968, A203847, A000204 (Lucas).

%Y Cf. A209447 (Pell variant).

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 04 2012