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A209447 a(n) = Pell(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.

%I A209447 #10 Jan 03 2018 17:54:56
%S A209447 1,12,72,60,1008,2088,2520,16224,73440,11820,513648,826704,1164240,
%T A209447 5621448,23265216,14041800,175149504,245524824,98791560,1590026160,
%U A209447 8061191712,3706940640,40272058656,64816900128,97801149600,487966581012,1596075244848,91744440540
%N A209447 a(n) = Pell(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 A209447 Compare g.f. to the Lambert series of A008653: 1 + 12*Sum_{n>=1} Chi(n,3)*n*x^n/(1-x^n).
%C A209447 Here Chi(n,3) = principal Dirichlet character of n modulo 3.
%H A209447 G. C. Greubel, <a href="/A209447/b209447.txt">Table of n, a(n) for n = 0..1000</a>
%F A209447 G.f.: 1 + 12*Sum_{n>=1} Pell(n)*Chi(n,3)*n*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).
%e A209447 G.f.: A(x) = 1 + 12*x + 72*x^2 + 60*x^3 + 1008*x^4 + 2088*x^5 + 2520*x^6 +...
%e A209447 where A(x) = 1 + 1*12*x + 2*36*x^2 + 5*12*x^3 + 12*84*x^4 + 29*72*x^5 + 70*36*x^6 +...+ Pell(n)*A008653(n)*x^n +...
%e A209447 The g.f. is also given by the identity:
%e A209447 A(x) = 1 + 12*( 1*1*x/(1-2*x-x^2) + 2*2*x^2/(1-6*x^2+x^4) + 12*4*x^4/(1-34*x^4+x^8) + 29*5*x^5/(1-82*x^5-x^10) + 169*7*x^7/(1-478*x^7-x^14) + 408*8*x^8/(1-1154*x^8-x^16)  +...).
%e A209447 The values of the Dirichlet character Chi(n,3) repeat [1,1,0, ...].
%t A209447 A008653[n_]:= If[n < 1, Boole[n == 0], 12*Sum[If[Mod[d, 3] > 0, d, 0], {d, Divisors@n}]]; Join[{1}, Table[Fibonacci[n, 2]*A008653[n], {n, 1, 1000}]] (* _G. C. Greubel_, Jan 02 2017 *)
%o A209447 (PARI) {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
%o A209447 {A002203(n)=Pell(n-1)+Pell(n+1)}
%o A209447 {a(n)=polcoeff(1 + 12*sum(m=1,n,Pell(m)*kronecker(m,3)^2*m*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
%o A209447 for(n=0,50,print1(a(n),", "))
%Y A209447 Cf. A008653, A205967, A209446, A209448, A204270, A000129 (Pell), A002203.
%K A209447 nonn
%O A209447 0,2
%A A209447 _Paul D. Hanna_, Mar 10 2012