cp's OEIS Frontend

A209443 a(n) = Pell(n)*A000118(n) for n>=1 with a(0)=1, where A000118(n) is the number of ways of writing n as a sum of 4 squares.

%I A209443 #8 Jan 03 2018 17:53:57
%S A209443 1,8,48,160,288,1392,6720,10816,9792,102440,342432,551136,1330560,
%T A209443 3747632,15510144,37444800,11299968,163683216,856193520,1060017440,
%U A209443 2303197632,9885175040,26848039104,43211266752,52160613120,325311054008,1064050163232,2446518414400
%N A209443 a(n) = Pell(n)*A000118(n) for n>=1 with a(0)=1, where A000118(n) is the number of ways of writing n as a sum of 4 squares.
%C A209443 Compare g.f. to the Lambert series of A000118: 1 + 8*Sum_{n>=1} n*x^n/(1+(-x)^n).
%H A209443 G. C. Greubel, <a href="/A209443/b209443.txt">Table of n, a(n) for n = 0..1000</a>
%F A209443 G.f.: 1 + 8*Sum_{n>=1} Pell(n)*n*x^n/(1 + A002203(n)*(-x)^n + (-1)^n*x^(2*n)).
%e A209443 G.f.: A(x) = 1 + 8*x + 48*x^2 + 160*x^3 + 288*x^4 + 1392*x^5 + 6720*x^6 +...
%e A209443 where A(x) = 1 + 1*8*x + 2*24*x^2 + 5*32*x^3 + 12*24*x^4 + 29*48*x^5 + 70*96*x^6 + 169*64*x^7 + 408*24*x^8 +...+ Pell(n)*A000118(n)*x^n +...
%e A209443 The g.f. is also given by the identity:
%e A209443 A(x) = 1 + 8*( 1*1*x/(1-2*x-x^2) + 2*2*x^2/(1+6*x^2+x^4) + 5*3*x^3/(1-14*x^3-x^6) + 12*4*x^4/(1+34*x^4+x^8) + 29*5*x^5/(1-82*x^5-x^10) + 70*6*x^6/(1+198*x^6+x^12) + 169*7*x^7/(1-478*x^7-x^14) +...).
%t A209443 A000118[n_]:= If[n < 1, Boole[n == 0], 8*Sum[If[Mod[d, 4] > 0, d, 0], {d, Divisors@n}]]; Join[{1}, Table[Fibonacci[n, 2]*A000118[n], {n, 1, 50}]] (* _G. C. Greubel_, Jan 02 2018 *)
%o A209443 (PARI) {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
%o A209443 {A002203(n)=Pell(n-1)+Pell(n+1)}
%o A209443 {a(n)=polcoeff(1+8*sum(m=1,n,Pell(m)*m*x^m/(1+A002203(m)*(-x)^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
%o A209443 for(n=0,30,print1(a(n),", "))
%Y A209443 Cf. A000118, A205963, A205508, A209444, A204270.
%K A209443 nonn
%O A209443 0,2
%A A209443 _Paul D. Hanna_, Mar 09 2012