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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. 3
1, 8, 48, 160, 288, 1392, 6720, 10816, 9792, 102440, 342432, 551136, 1330560, 3747632, 15510144, 37444800, 11299968, 163683216, 856193520, 1060017440, 2303197632, 9885175040, 26848039104, 43211266752, 52160613120, 325311054008, 1064050163232, 2446518414400 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Compare g.f. to the Lambert series of A000118: 1 + 8*Sum_{n>=1} n*x^n/(1+(-x)^n).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

G.f.: 1 + 8*Sum_{n>=1} Pell(n)*n*x^n/(1 + A002203(n)*(-x)^n + (-1)^n*x^(2*n)).

EXAMPLE

G.f.: A(x) = 1 + 8*x + 48*x^2 + 160*x^3 + 288*x^4 + 1392*x^5 + 6720*x^6 +...

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 +...

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

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) +...).

MATHEMATICA

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 *)

PROG

(PARI) {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)}

{A002203(n)=Pell(n-1)+Pell(n+1)}

{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)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A000118, A205963, A205508, A209444, A204270.

Sequence in context: A222816 A280056 A035471 * A072819 A190317 A073912

Adjacent sequences:  A209440 A209441 A209442 * A209444 A209445 A209446

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Mar 09 2012

STATUS

approved

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Last modified November 17 00:02 EST 2018. Contains 317275 sequences. (Running on oeis4.)