A204270 -id:A204270 - 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.

A209446 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 + xy + y^2 = n.

Original entry on oeis.org

1, 6, 0, 30, 72, 0, 0, 2028, 0, 5910, 0, 0, 83160, 401532, 0, 0, 2824992, 0, 0, 79501308, 0, 463367580, 0, 0, 0, 7870428726, 0, 45872220270, 221490672624, 0, 0, 3116610274188, 0, 0, 0, 0, 127800022137480, 617073093431772, 0, 3596565555708780, 0, 0, 0, 122177355889216668

Offset: 0

Paul D. Hanna, Mar 10 2012

nonn

Compare g.f. to the Lambert series of A004016: 1 + 6*Sum_{n>=1} Kronecker(n,3)*x^n/(1 - x^n).

			G.f.: A(x) = 1 + 6*x + 30*x^3 + 72*x^4 + 2028*x^7 + 5910*x^9 + 83160*x^12 + ...
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 + ...
The g.f. is also given by the identity:
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) + ...).
The values of the symbol Kronecker(n,3) repeat [1, -1, 0, ...].

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

Cf. A004016, A205966, A209445, A209447, A204270, A000129 (Pell), A002203.

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 *)

{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 + 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)}
for(n=0,60,print1(a(n),", "))

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).

A204271 a(n) = sigma(n)*Pell(n), where sigma(n) = A000203(n), the sum of divisors of n.

Original entry on oeis.org

1, 6, 20, 84, 174, 840, 1352, 6120, 12805, 42804, 68892, 388080, 468454, 1938768, 4680600, 14595792, 20460402, 107024190, 132502180, 671765976, 1235646880, 3356004888, 5401408344, 32600383200, 40663881751, 133006270404, 305814801800

Offset: 1

Paul D. Hanna, Jan 14 2012

nonn

Compare g.f. to the Lambert series identity: Sum_{n>=1} n*x^n/(1-x^n) = Sum_{n>=1} sigma(n)*x^n.

			G.f.: A(x) = x + 6*x^2 + 20*x^3 + 84*x^4 + 174*x^5 + 840*x^6 + 1352*x^7 +...
where A(x) = x/(1-2*x-x^2) + 2*2*x^2/(1-6*x^2+x^4) + 3*5*x^3/(1-14*x^3-x^6) + 4*12*x^4/(1-34*x^4+x^8) + 5*29*x^5/(1-82*x^5-x^10) + 6*70*x^6/(1-198*x^6+x^12) +...+ n*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) +...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000203 (sigma), A000129, A203848, A204270, A204272, A204273, A204274, A204275, A002203, A000045.

Table[DivisorSigma[1, n] Fibonacci[n, 2], {n, 1, 50}] (* G. C. Greubel, Jan 05 2018 *)

/* 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)=sigma(n)*Pell(n)}
```

{a(n)=polcoeff(sum(m=1,n,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} n*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma(n)*Pell(n)*x^n, where Pell(n) = A000129(n) and A002203 is the companion Pell numbers.

A204272 a(n) = sigma_2(n)*Pell(n), where sigma_2(n) = A001157(n), the sum of squares of divisors of n.

Original entry on oeis.org

1, 10, 50, 252, 754, 3500, 8450, 34680, 89635, 309140, 700402, 2910600, 5688370, 20195500, 50706500, 160553712, 329639810, 1248615550, 2398289458, 8732957688, 19306982500, 56865638380, 119281100930, 461838762000, 853941516771

Offset: 1

Paul D. Hanna, Jan 14 2012

nonn

Compare g.f. to the Lambert series identity: Sum_{n>=1} n^2*x^n/(1-x^n) = Sum_{n>=1} sigma_2(n)*x^n.

			G.f.: A(x) = x + 10*x^2 + 50*x^3 + 252*x^4 + 754*x^5 + 3500*x^6 +...
where A(x) = x/(1-2*x-x^2) + 2^2*2*x^2/(1-6*x^2+x^4) + 3^2*5*x^3/(1-14*x^3-x^6) + 4^2*12*x^4/(1-34*x^4+x^8) + 5^2*29*x^5/(1-82*x^5-x^10) + 6^2*70*x^6/(1-198*x^6+x^12) +...+ n^2*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) +...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A203849, A204270, A204271, A204273, A204274, A204275, A001157 (sigma_2), A002203, A000045.

With[{nn=30},Times@@@Thread[{Rest[LinearRecurrence[{2,1},{0,1},nn+1]], DivisorSigma[ 2,Range[nn]]}]] (* Harvey P. Dale, Oct 21 2015 *)

/* 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)=sigma(n,2)*Pell(n)}
```

{a(n)=polcoeff(sum(m=1,n,m^2*Pell(m)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}

G.f.: Sum_{n>=1} n^2*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_2(n)*Pell(n)*x^n, where Pell(n) = A000129(n) and A002203 is the companion Pell numbers.

A204273 a(n) = sigma_3(n)*Pell(n), where sigma_3(n) = A001158(n), the sum of cubes of divisors of n.

Original entry on oeis.org

1, 18, 140, 876, 3654, 17640, 58136, 238680, 745645, 2696652, 7647012, 28329840, 73547278, 250101072, 688048200, 2203964592, 5585689746, 18696302730, 45448247740, 147116748744, 371929710880, 1117549627704, 2738514030408, 8899904613600

Offset: 1

Paul D. Hanna, Jan 14 2012

nonn

Compare g.f. to the Lambert series identity: Sum_{n>=1} n^3*x^n/(1-x^n) = Sum_{n>=1} sigma_3(n)*x^n.

			G.f.: A(x) = x + 18*x^2 + 140*x^3 + 876*x^4 + 3654*x^5 + 17640*x^6 + ...
where A(x) = x/(1-2*x-x^2) + 2^3*2*x^2/(1-6*x^2+x^4) + 3^3*5*x^3/(1-14*x^3-x^6) + 4^3*12*x^4/(1-34*x^4+x^8) + 5^3*29*x^5/(1-82*x^5-x^10) + 6^3*70*x^6/(1-198*x^6+x^12) + ... + n^3*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A203838, A204270, A204271, A204272, A204274, A204275, A001158 (sigma_3), A002203, A000045.

Table[DivisorSigma[3, n] Fibonacci[n, 2], {n, 1, 50}] (* G. C. Greubel, Jan 05 2018 *)

/* 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)=sigma(n,3)*Pell(n)}
```

{a(n)=polcoeff(sum(m=1,n,m^3*Pell(m)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}

G.f.: Sum_{n>=1} n^3*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_3(n)*Pell(n)*x^n, where Pell(n) = A000129(n) and A002203 is the companion Pell numbers.

A204275 G.f.: Product_{n>=1} (1 + A002203(n)x^n + (-1)^nx^(2*n)) where A002203 is the companion Pell numbers.

Original entry on oeis.org

1, 2, 5, 26, 57, 222, 698, 2096, 6038, 19730, 58915, 169952, 516024, 1484958, 4397513, 13029558, 37094682, 106442928, 311875984, 879620854, 2522107990, 7229956352, 20398904648, 57543374566, 163053304047, 457604617760, 1283583473614, 3606627675050

Offset: 0

Paul D. Hanna, Jan 13 2012

nonn

Analog to Euler's identity: Product_{n>=1} (1+x^n) = Product_{n>=1} 1/(1-x^(2*n-1)), which is the g.f. for the number of partitions of distinct parts.

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 26*x^3 + 57*x^4 + 222*x^5 + 698*x^6 +...
where A(x) = (1+2*x-x^2) * (1+6*x^2+x^4) * (1+14*x^3-x^6) * (1+34*x^4+x^8) * (1+82*x^5-x^10) * (1+198*x^6+x^12) *...* (1 + A002203(n)*x^n + (-1)^n*x^(2*n)) *...
and 1/A(x) = (1-2*x-x^2) * (1-14*x^3-x^6) * (1-82*x^5-x^10) * (1-478*x^7-x^14) * (1-2786*x^9-x^18) * (1-16238*x^11-x^22) *...* (1 - A002203(2*n-1)*x^(2*n-1) + (-1)^n*x^(4*n-2)) *...
Also, the logarithm of the g.f. equals the series:
log(A(x)) = 1*2*x + 1*6*x^2/2 + 4*14*x^3/3 + 1*34*x^4/4 + 6*82*x^5/5 + 4*198*x^6/6 + 8*478*x^7/7 + 1*1154*x^8/8 +...+ A000593(n)*A002203(n)*x^n/n +...
The companion Pell numbers (starting at offset 1) begin:
A002203 = [2,6,14,34,82,198,478,1154,2786,6726,16238,...]
and form the logarithm of a g.f. for Pell numbers:
log(1/(1-2*x-x^2)) = 2*x + 6*x^2/2 + 14*x^3/3 + 34*x^4/4 + 82*x^5/5 +...

Eric Weisstein's World of Mathematics, Euler Identity.

Cf. A203801, A204270, A000129 (Pell), A002203 (companion Pell), A000593.

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

{a(n)=polcoeff(prod(k=1,n,1+A002203(k)*x^k+(-1)^k*x^(2*k) +x*O(x^n)),n)}

{a(n)=polcoeff(1/prod(k=1,n,1-A002203(2*k-1)*x^(2*k-1)-x^(4*k-2) +x*O(x^n)),n)}

/* Exponential form using sum of odd divisors of n: */
{A000593(n)=if(n<1, 0, sumdiv(n, d, (-1)^(d+1)*n/d))}
{a(n)=polcoeff(exp(sum(k=1, n, A000593(k)*A002203(k)*x^k/k)+x*O(x^n)), n)}

G.f.: Product_{n>=1} 1/(1 - A002203(2*n-1)*x^(2*n-1) + (-1)^n*x^(4*n-2)).
G.f.: exp( Sum_{n>=1} A000593(n) * A002203(n) * x^n/n ) where A000593(n) = sum of odd divisors of n.
a(n) = (1/n)*Sum_{k=1..n} A000593(k) * A002203(k)*a(n-k) for n>0, with a(0) = 1.

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.

Original entry on oeis.org

1, 12, 72, 60, 1008, 2088, 2520, 16224, 73440, 11820, 513648, 826704, 1164240, 5621448, 23265216, 14041800, 175149504, 245524824, 98791560, 1590026160, 8061191712, 3706940640, 40272058656, 64816900128, 97801149600, 487966581012, 1596075244848, 91744440540

Offset: 0

Paul D. Hanna, Mar 10 2012

nonn

Compare g.f. to the Lambert series of A008653: 1 + 12*Sum_{n>=1} Chi(n,3)*n*x^n/(1-x^n).

Here Chi(n,3) = principal Dirichlet character of n modulo 3.

			G.f.: A(x) = 1 + 12*x + 72*x^2 + 60*x^3 + 1008*x^4 + 2088*x^5 + 2520*x^6 +...
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 +...
The g.f. is also given by the identity:
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)  +...).
The values of the Dirichlet character Chi(n,3) repeat [1,1,0, ...].

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

Cf. A008653, A205967, A209446, A209448, A204270, A000129 (Pell), A002203.

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 *)

{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 + 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)}
for(n=0,50,print1(a(n),", "))

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).

A209449 a(n) = Pell(n)*A113973(n) for n>=1, with a(0)=1, where A113973 lists the coefficients in phi(x^3)^3/phi(x) and phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 8, -10, 24, 0, 280, -676, 1632, -1970, 0, 0, 27720, -133844, 646256, 0, 941664, 0, 10976840, -26500436, 0, -154455860, 0, 0, 2173358880, -2623476242, 25334527696, -15290740090, 73830224208, 0, 0, -1038870091396, 2508054264192, 0, 0, 0, 42600007379160

Offset: 0

Paul D. Hanna, Mar 10 2012

sign

Compare g.f. to the Lambert series of A113973:

1 - 2*Sum_{n>=1} Kronecker(n,3)*x^n/(1 - (-x)^n).

			G.f.: A(x) = 1 - 2*x + 8*x^2 - 10*x^3 + 24*x^4 + 280*x^6 - 676*x^7 +...
where A(x) = 1 - 1*2*x + 2*4*x^2 - 5*2*x^3 + 12*2*x^4 + 70*4*x^6 - 169*4*x^7 + 408*4*x^8 +...+ Pell(n)*A113973(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 - 2*( 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) - 408*x^8/(1-1154*x^8+x^16) +...).
The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].

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

Cf. A113973, A205969, A209446, A209448, A209450, A204270, A000129 (Pell), A002203.

A113973:= CoefficientList[Series[EllipticTheta[3, 0, q^3]^3 /EllipticTheta[3, 0, q], {q, 0, 60}], q]; Table[If[n == 0, 1, Fibonacci[n, 2]*A113973[[n + 1]]], {n, 0, 50}] (* G. C. Greubel, Dec 03 2017 *)

{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 - 2*sum(m=1,n,Pell(m)*kronecker(m,3)*x^m/(1-A002203(m)*(-x)^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,60,print1(a(n),", "))

G.f.: 1 - 2*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).

A209444 a(n) = Pell(n)*A000143(n) for n>=1 with a(0)=1, where A000143(n) is the number of ways of writing n as a sum of 8 squares.

Original entry on oeis.org

1, 16, 224, 2240, 13632, 58464, 219520, 930176, 3805824, 11930320, 33558336, 122352192, 440858880, 1176756448, 3112368896, 11008771200, 35248366848, 89371035936, 232665100640, 727171963840, 2289378446208, 5950875374080, 13907284255872, 43816224486528

Offset: 0

Paul D. Hanna, Mar 09 2012

nonn

Compare g.f. to the Lambert series of A000143: 1 + 16*Sum_{n>=1} n^3*x^n/(1 - (-x)^n).

			G.f.: A(x) = 1 + 16*x + 112*x^2 + 896*x^3 + 3408*x^4 + 10080*x^5 +...
where A(x) = 1 + 1*16*x + 2*112*x^2 + 5*448*x^3 + 12*1136*x^4 + 29*2016*x^5 + 70*3136*x^6 + 169*5504*x^7 + 408*9328*x^8 +...+ Pell(n)*A000143(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 16*( 1*1*x/(1+2*x-x^2) + 2*8*x^2/(1-6*x^2+x^4) + 5*27*x^3/(1+14*x^3-x^6) + 12*64*x^4/(1-34*x^4+x^8) + 29*125*x^5/(1+82*x^5-x^10) + 70*216*x^6/(1-198*x^6+x^12) + 169*343*x^7/(1+478*x^7-x^14) +...).

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

Cf. A000143, A205964, A205508, A209443, A204270.

A000143:= Table[SquaresR[8, n], {n, 0, 200}]; Join[{1}, Table[Fibonacci[n, 2]*A000143[[n + 1]], {n, 1, 50}]] (* G. C. Greubel, Jan 02 2018 *)

{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+16*sum(m=1,n,Pell(m)*m^3*x^m/(1-A002203(m)*(-x)^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
for(n=0,31,print1(a(n),", "))

G.f.: 1 + 16*Sum_{n>=1} Pell(n)*n^3*x^n/(1 - A002203(n)*(-x)^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).

A209448 a(n) = Pell(n)A008655(n) for n>=1, with a(0)=1, where A008655 lists the coefficients in (theta_3(x)theta_3(3x)+theta_2(x)theta_2(3*x))^4.

Original entry on oeis.org

1, 24, 432, 4440, 21024, 87696, 559440, 1395264, 5728320, 23852760, 64719648, 183528288, 898460640, 1765134672, 6002425728, 21820957200, 52895150208, 134056553904, 598084104240, 1090757945760, 3530801969856, 11795485116480, 26821191064896, 65724336729792

Offset: 0

Paul D. Hanna, Mar 10 2012

nonn

Compare g.f. to the Lambert series of A008655:

1 + Sum_{n>=1} 24*n^3*x^n/(1-x^n) + 8*(3*n)^3*x^(3*n)/(1-x^(3*n)).

			G.f.: A(x) = 1 + 24*x + 432*x^2 + 4440*x^3 + 21024*x^4 + 87696*x^5 +...
where A(x) = 1 + 1*24*x + 2*216*x^2 + 5*888*x^3 + 12*1752*x^4 + 29*3024*x^5 +...+ Pell(n)*A008655(n)*x^n +...

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

Cf. A008655, A205968, A209447, A209449, A204270, A000129 (Pell), A002203.

A008655[n_]:= SeriesCoefficient[((EllipticTheta[3, 0, q]^3 + EllipticTheta[3, Pi/3, q]^3 + EllipticTheta[3, 2 Pi/3, q]^3)^4/(3* EllipticTheta[3, 0, q^3])^4), {q, 0, n}]; b:= Table[A008655[n], {n, 0, 102}][[1 ;; ;; 2]]; Join[{1}, Table[Fibonacci[n, 2]*b[[n + 1]], {n, 1, 50}]] (* G. C. Greubel, Jan 26 2018 *)

{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 + sum(m=1,n, 24*Pell(m)*m^3*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n)) + 8*Pell(3*m)*(3*m)^3*x^(3*m)/(1-A002203(3*m)*x^(3*m)+(-1)^m*x^(6*m) +x*O(x^n))  ),n)}
for(n=0,40,print1(a(n),", "))

G.f.: 1 + Sum_{n>=1} 24*Pell(n)*n^3*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) + 8*Pell(3*n)*(3*n)^3*x^(3*n)/(1 - A002203(3*n)*x^(3*n) + (-1)^n*x^(6*n)), where A002203(n) = Pell(n-1) + Pell(n+1).

cp's OEIS Frontend

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

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A209446 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.

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A204271 a(n) = sigma(n)*Pell(n), where sigma(n) = A000203(n), the sum of divisors of n.

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A204272 a(n) = sigma_2(n)*Pell(n), where sigma_2(n) = A001157(n), the sum of squares of divisors of n.

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A204273 a(n) = sigma_3(n)*Pell(n), where sigma_3(n) = A001158(n), the sum of cubes of divisors of n.

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A204275 G.f.: Product_{n>=1} (1 + A002203(n)*x^n + (-1)^n*x^(2*n)) where A002203 is the companion Pell numbers.

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A209446 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 + xy + y^2 = n.

A204275 G.f.: Product_{n>=1} (1 + A002203(n)x^n + (-1)^nx^(2*n)) where A002203 is the companion Pell numbers.

A209448 a(n) = Pell(n)A008655(n) for n>=1, with a(0)=1, where A008655 lists the coefficients in (theta_3(x)theta_3(3x)+theta_2(x)theta_2(3*x))^4.