|
%I #13 Sep 06 2018 02:43:51
%S 1,6,0,30,72,0,0,2028,0,5910,0,0,83160,401532,0,0,2824992,0,0,
%T 79501308,0,463367580,0,0,0,7870428726,0,45872220270,221490672624,0,0,
%U 3116610274188,0,0,0,0,127800022137480,617073093431772,0,3596565555708780,0,0,0,122177355889216668
%N a(n) = Pell(n)*A004016(n) for n >= 1, with a(0)=1, where A004016(n) is the number of integer solutions (x,y) to x^2 + x*y + y^2 = n.
%C Compare g.f. to the Lambert series of A004016: 1 + 6*Sum_{n>=1} Kronecker(n,3)*x^n/(1 - x^n).
%H G. C. Greubel, <a href="/A209446/b209446.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: 1 + 6*Sum_{n>=1} Pell(n)*Kronecker(n,3)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).
%e G.f.: A(x) = 1 + 6*x + 30*x^3 + 72*x^4 + 2028*x^7 + 5910*x^9 + 83160*x^12 + ...
%e where A(x) = 1 + 1*6*x + 5*6*x^3 + 12*6*x^4 + 169*12*x^7 + 985*6*x^9 + 13860*6*x^12 + ... + Pell(n)*A004016(n)*x^n + ...
%e The g.f. is also given by the identity:
%e A(x) = 1 + 6*( 1*x/(1-2*x-x^2) - 2*x^2/(1-6*x^2+x^4) + 12*x^4/(1-34*x^4+x^8) - 29*x^5/(1-82*x^5-x^10) + 169*x^7/(1-478*x^7-x^14) + ...).
%e The values of the symbol Kronecker(n,3) repeat [1, -1, 0, ...].
%t A004016[n_]:= If[n < 1, Boole[n == 0], 6 DivisorSum[n, KroneckerSymbol[#, 3] &]]; Join[{1}, Table[Fibonacci[n, 2]*A004016[n], {n, 1, 50}]] (* _G. C. Greubel_, Jan 02 2018 *)
%o (PARI) {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
%o {A002203(n)=Pell(n-1)+Pell(n+1)}
%o {a(n)=polcoeff(1 + 6*sum(m=1,n,kronecker(m,3)*Pell(m)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
%o for(n=0,60,print1(a(n),", "))
%Y Cf. A004016, A205966, A209445, A209447, A204270, A000129 (Pell), A002203.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Mar 10 2012
|
