A204327 -id:A204327 - cp's OEIS frontend

A204274 G.f.: Sum_{n>=1} Pell(n^2)*x^(n^2).

1, 0, 0, 12, 0, 0, 0, 0, 985, 0, 0, 0, 0, 0, 0, 470832, 0, 0, 0, 0, 0, 0, 0, 0, 1311738121, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21300003689580, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2015874949414289041, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Paul D. Hanna, Jan 14 2012

nonn

Compare g.f. to the Lambert series identity: Sum_{n>=1} lambda(n)*x^n/(1-x^n) = Sum_{n>=1} x^(n^2); Liouville's function lambda(n) = (-1)^k, where k is number of primes dividing n (counted with multiplicity).

			G.f.: A(x) = x + 12*x^4 + 985*x^9 + 470832*x^16 + 1311738121*x^25 +...
where A(x) = x/(1-2*x-x^2) + (-1)*2*x^2/(1-6*x^2+x^4) + (-1)*5*x^3/(1-14*x^3-x^6) + (+1)*12*x^4/(1-34*x^4+x^8) + (-1)*29*x^5/(1-82*x^5-x^10) + (+1)*70*x^6/(1-198*x^6+x^12) +...+ lambda(n)*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) +...

Robert Israel, Table of n, a(n) for n = 1..2500

Cf. A204327, A204060, A204270, A204271, A204272, A204273, A204275, A008836 (lambda), A002203, A000045.

pell:= gfun:-rectoproc({a(0)=0,a(1)=1,a(n)=2*a(n-1)+a(n-2)},a(n),remember):
seq(`if`(issqr(n),pell(n),0), n=1..100); # Robert Israel, Nov 24 2015

CoefficientList[Sum[Fibonacci[n^2, 2] x^n^2/x, {n, 1, 8}], x] (* Jean-François Alcover, Mar 25 2019 *)

/* Subroutines used in PARI programs below: */
{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)}
{A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}

PARI
```
{a(n)=issquare(n)*Pell(n)}
```

{lambda(n)=local(F=factor(n));(-1)^sum(i=1,matsize(F)[1],F[i,2])}
{a(n)=polcoeff(sum(m=1,n,lambda(m)*Pell(m)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}

G.f.: Sum_{n>=1} lambda(n)*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where lambda(n) = A008836(n), Pell(n) = A000129(n) and A002203 is the companion Pell numbers.

A211891 G.f.: exp( Sum_{n>=1} 2 * Pell(n^2) * x^n/n ), where Pell(n) = A000129(n).

Original entry on oeis.org

1, 2, 14, 682, 236826, 525175434, 7101054148862, 575978478770467714, 277997363115795461721154, 794462328877965002894838885122, 13398419999037765629218732004567606814, 1330302023374557034879527995005574743144202826

Offset: 0

Paul D. Hanna, Apr 24 2012

nonn

Given g.f. A(x), note that A(x)^(1/2) is not an integer series.

			G.f.: A(x) = 1 + 2*x + 14*x^2 + 682*x^3 + 236826*x^4 + 525175434*x^5 +...
such that
log(A(x))/2 = x + 12*x^2/2 + 985*x^3/3 + 470832*x^4/4 + 1311738121*x^5/5 + 21300003689580*x^6/6 + 2015874949414289041*x^7/7 +...+ Pell(n^2)*x^n/n +...
Pell numbers begin:
A000129 = [1,2,5,12,29,70,169,408,985,2378,5741,13860,33461,...].

Cf. A208056, A211892, A000129 (Pell), A204327 (Pell(n^2)).

{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, 2*Pell(k^2)*x^k/k)+x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))

A380083 a(n) = Pell(n^2)/Pell(n).

Original entry on oeis.org

1, 6, 197, 39236, 45232349, 304285766994, 11928254138546089, 2725453049877127789064, 3629520789795568149638626009, 28171611459441395148628640333550174, 1274457582507820938168220698796661580252461, 336039604487720392926819615640785342048933644491212

Offset: 1

Seiichi Manyama, May 08 2025

Main diagonal of A383742.

Cf. A000129, A002203, A051294, A204327.

a:= n-> (f->f(n^2)/f(n))(k->(<<2|1>, <1|0>>^k)[1, 2]):
seq(a(n), n=1..12);  # Alois P. Heinz, May 08 2025

a[n_] := Fibonacci[n^2, 2]/Fibonacci[n, 2]; Array[a, 12] (* Amiram Eldar, May 08 2025 *)

pell(n) = ([2, 1; 1, 0]^n)[2, 1];
a(n) = pell(n^2)/pell(n);

a(n) = A204327(n)/A000129(n).

a(n) = [x^n] x/(1 - A002203(n)*x + (-1)^n*x^2).

cp's OEIS Frontend

A204274 G.f.: Sum_{n>=1} Pell(n^2)*x^(n^2).

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A211891 G.f.: exp( Sum_{n>=1} 2 * Pell(n^2) * x^n/n ), where Pell(n) = A000129(n).

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A380083 a(n) = Pell(n^2)/Pell(n).

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Mathematica

PARI

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