cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A209450 a(n) = Pell(n)*A132973(n) for n>=1, with a(0)=1, where A132973 lists the coefficients in psi(-q)^3/psi(-q^3) and where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -3, 6, -15, 36, 0, 210, -1014, 1224, -2955, 0, 0, 41580, -200766, 484692, 0, 1412496, 0, 8232630, -39750654, 0, -231683790, 0, 0, 1630019160, -3935214363, 19000895772, -22936110135, 110745336312, 0, 0, -1558305137094, 1881040698144, 0, 0, 0, 63900011068740
Offset: 0

Views

Author

Paul D. Hanna, Mar 10 2012

Keywords

Comments

Compare g.f. to the Lambert series of A132973: 1 - 3*Sum_{n>=0} x^(6*n+1)/(1+x^(6*n+1)) - x^(6*n+5)/(1+x^(6*n+5)).

Examples

			G.f.: A(x) = 1 - 3*x + 6*x^2 - 15*x^3 + 36*x^4 + 210*x^6 - 1014*x^7 +...
where A(x) = 1 - 1*3*x + 2*3*x^2 - 5*3*x^3 + 12*3*x^4 + 70*3*x^6 - 169*6*x^7 + 408*3*x^8 +...+ Pell(n)*A132973(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 - 3*( 1*x/(1+2*x-x^2) - 29*x^5/(1+82*x^5-x^10) + 169*x^7/(1+478*x^7-x^14) - 5741*x^11/(1+16238*x^11-x^22) + 33461*x^13/(1+94642*x^13-x^26) - 1136689*x^17/(1+3215042*x^17-x^34) +...).

Links

Crossrefs

Cf. A132973, A205970, A209449, A209451, A204270, A000129 (Pell), A002203.

Programs

  • Mathematica
    A132973[n_]:= SeriesCoefficient[EllipticTheta[2, Pi/4, q^(1/2)]^3/EllipticTheta[2, Pi/4, q^(3/2)]/2, {q, 0, n}]; Join[{1}, Table[ Fibonacci[n, 2]*A132973[n],{n,1,50}]] (* G. C. Greubel, Jan 02 2018 *)
  • 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 - 3*sum(m=0,n, Pell(6*m+1)*x^(6*m+1)/(1+A002203(6*m+1)*x^(6*m+1)-x^(12*m+2) +x*O(x^n)) - Pell(6*m+5)*x^(6*m+5)/(1+A002203(6*m+5)*x^(6*m+5)-x^(12*m+10) +x*O(x^n)) ),n)}
for(n=0,61,print1(a(n),", "))

Formula

G.f.: 1 - 3*Sum_{n>=0} Pell(6*n+1)*x^(6*n+1)/(1+A002203(6*n+1)*x^(6*n+1)-x^(12*n+2)) - Pell(6*n+5)*x^(6*n+5)/(1+A002203(6*n+5)*x^(6*n+5)-x^(12*n+10)), where A002203(n) = Pell(n-1) + Pell(n+1).

A209451 a(n) = Pell(n)*A034896(n) for n >= 1, with a(0)=1, where A034896 lists the number of solutions to a^2 + b^2 + 3*c^2 + 3*d^2 = n.

Original entry on oeis.org

1, 4, 8, 20, 240, 696, 280, 5408, 21216, 3940, 57072, 275568, 277200, 1873816, 2585024, 4680600, 54616512, 81841608, 10976840, 530008720, 1919331360, 1235646880, 4474673184, 21605633376, 28253665440, 162655527004, 177341693872, 30581480180, 2953208968320
Offset: 0

Views

Author

Paul D. Hanna, Mar 10 2012

Keywords

Comments

Compare g.f. to the Lambert series of A034896:
1 + 4*Sum_{n>=1} Chi(n,3)*n*x^n/(1 - (-x)^n).
Here Chi(n,3) = principal Dirichlet character modulo 3.

Examples

			G.f.: A(x) = 1 + 4*x + 8*x^2 + 20*x^3 + 240*x^4 + 696*x^5 + 280*x^6 + ...
where A(x) = 1 + 1*4*x + 2*4*x^2 + 5*4*x^3 + 12*20*x^4 + 29*24*x^5 + 70*4*x^6 + ... + Pell(n)*A034896(n)*x^n + ...
The g.f. is also given by the identity:
A(x) = 1 + 4*( 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,...].

Links

Crossrefs

Cf. A034896, A205971, A205884, A209447, A209450, A209452, A204270, A000129 (Pell), A002203.

Programs

  • Mathematica
    A034896[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^3])^2, {q, 0, n}]; Join[{1}, Table[Fibonacci[n, 2]*A034896[n], {n, 1, 50}]] (* G. C. Greubel, Dec 24 2017 *)
  • 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 + 4*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=0,61,print1(a(n),", "))

Formula

G.f.: 1 + 4*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).

A209452 a(n) = Pell(n)*A122859(n) for n>=1, with a(0)=1, where A122859 lists the coefficients in phi(-q)^3/phi(-q^3) and phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -6, 24, -30, -72, 0, 840, -2028, 4896, -5910, 0, 0, -83160, -401532, 1938768, 0, -2824992, 0, 32930520, -79501308, 0, -463367580, 0, 0, 6520076640, -7870428726, 76003583088, -45872220270, -221490672624, 0, 0, -3116610274188, 7524162792576, 0, 0, 0, -127800022137480
Offset: 0

Views

Author

Paul D. Hanna, Mar 10 2012

Keywords

Comments

Compare the g.f. to the Lambert series of A122859: 1 - 6*Sum_{n>=1} Kronecker(n,3)*x^n/(1+x^n).

Examples

			G.f.: A(x) = 1 - 6*x + 24*x^2 - 30*x^3 - 72*x^4 + 840*x^6 - 2028*x^7 + ...
where A(x) = 1 - 1*6*x + 2*12*x^2 - 5*6*x^3 - 12*6*x^4 + 70*12*x^6 - 169*12*x^7 + 408*12*x^8 - 985*6*x^9 + ... + Pell(n)*A122859(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) - 408*x^8/(1+1154*x^8+x^16) + ...).
The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].

Links

Crossrefs

Cf. A122859, A205972, A209451, A209453, A209446, A209449, A204270, A000129 (Pell), A002203.

Programs

  • Mathematica
    A122859[n_]:= SeriesCoefficient[EllipticTheta[4, 0, q]^3/EllipticTheta[4, 0, q^3], {q, 0, n}]; Join[{1}, Table[Fibonacci[n, 2]*A122859[n], {n, 1, 50}]] (* G. C. Greubel, Jan 02 2017 *)
  • 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 - 6*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=0,40,print1(a(n),", "))

Formula

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).

A209453 a(n) = Pell(n)*A109041(n) for n>=1, with a(0)=1, where A109041 lists the coefficients in eta(q)^9/eta(q^3)^3.

Original entry on oeis.org

1, -9, 54, -45, -1404, 6264, 1890, -76050, 187272, -8865, -1540944, 6200280, -1621620, -51195330, 109055700, 42125400, -868685040, 2946297888, 74093670, -21584605122, 44912353824, -17376284250, -302040439920, 1069478852112, 249392931480, -7095191496489
Offset: 0

Views

Author

Paul D. Hanna, Mar 10 2012

Keywords

Comments

Compare the g.f. to the Lambert series of A109041:
1 - 9*Sum_{n>=1} Kronecker(n,3)*n^2*x^n/(1-x^n).

Examples

			G.f.: A(x) = 1 - 9*x + 54*x^2 - 45*x^3 - 1404*x^4 + 6264*x^5 + 1890*x^6 +...
where A(x) = 1 - 1*9*x + 2*27*x^2 - 5*9*x^3 - 12*117*x^4 + 29*216*x^5 + 70*27*x^6 - 169*450*x^7 + 408*459*x^8 +...+ Pell(n)*A109041(n)*^n +...
The g.f. is also given by the identity:
A(x) = 1 - 9*( 1*1*x/(1-2*x-x^2) - 2*4*x^2/(1-6*x^2+x^4) + 12*16*x^4/(1-34*x^4+x^8) - 29*25*x^5/(1-82*x^5-x^10) + 169*49*x^7/(1-478*x^7-x^14) - 408*64*x^8/(1-1154*x^8+x^16) +...).
The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].

Links

Crossrefs

Cf. A109041, A205973, A209452, A209454, A204270, A000129 (Pell), A002203.

Programs

  • Mathematica
    A109041[n_]:= If[n < 1, Boole[n == 0], -9 DivisorSum[n, #^2 KroneckerSymbol[-3, #] &]]; Join[{1}, Table[Fibonacci[n, 2]*A109041[n], {n, 1, 50}]] (* G. C. Greubel, Jan 02 2018 *)
  • 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 - 9*sum(m=1,n,Pell(m)*kronecker(m,3)*m^2*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,40,print1(a(n),", "))

Formula

G.f.: 1 - 9*Sum_{n>=1} Pell(n)*Kronecker(n,3)*n^2*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).

A209454 a(n) = Pell(n)*A033719(n) for n>=1, with a(0)=1, where A033719 lists the coefficients in theta_3(q)*theta_3(q^7).

Original entry on oeis.org

1, 2, 0, 0, 24, 0, 0, 338, 1632, 1970, 0, 22964, 0, 0, 0, 0, 2824992, 0, 0, 0, 0, 0, 0, 900234724, 0, 2623476242, 0, 0, 36915112104, 178241928596, 0, 0, 5016108528384, 0, 0, 0, 42600007379160, 205691031143924, 0, 0, 0, 0, 0, 40725785296405556, 98320743200877072, 0, 0, 0, 0
Offset: 0

Views

Author

Paul D. Hanna, Mar 10 2012

Keywords

Comments

Compare g.f. to 1 + 2*Sum_{n>=1} Kronecker(n,7)*x^n/(1-(-x)^n) (the Lambert series of A033719).

Examples

			G.f.: A(x) = 1 + 2*x + 24*x^4 + 338*x^7 + 1632*x^8 + 1970*x^9 + 22964*x^11 +...
where A(x) = 1 + 1*2*x + 12*2*x^4 + 169*2*x^7 + 408*4*x^8 + 985*2*x^9 + 5741*4*x^11 + 470832*6*x^16 + 225058681*4*x^23 +...+ Pell(n)*A033719(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) - 5*x^3/(1+14*x^3-x^6) + 12*x^4/(1-34*x^4+x^8) - 29*x^5/(1+82*x^5-x^10) - 70*x^6/(1-198*x^6+x^12) + 0*169*13*x^7/(1+478*x^7-x^14) +...).
The values of the symbol Kronecker(n,7) repeat [1,1,-1,1,-1,-1,0, ...].

Links

Crossrefs

Cf. A033719, A205974, A209453, A209455, A204270, A000129 (Pell), A002203.

Programs

  • Mathematica
    A033719[n_]:= SeriesCoefficient[EllipticTheta[3, 0, x] EllipticTheta[3, 0, x^7], {x, 0, n}]; Join[{1}, Table[Fibonacci[n, 2]*A033719[n], {n, 1, 50}]] (* G. C. Greubel, Jan 02 2017 *)
  • 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 + 2*sum(m=1,n,Pell(m)*kronecker(m,7)*x^m/(1-A002203(m)*(-x)^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,50,print1(a(n),", "))

Formula

G.f.: 1 + 2*Sum_{n>=1} Pell(n)*Kronecker(n,7)*x^n/(1 - A002203(n)*(-x)^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).

A209455 a(n) = Pell(n)*A002652(n) for n>=1, with a(0)=1, where A002652 lists the coefficients in theta series of Kleinian lattice Z[(-1+sqrt(-7))/2].

Original entry on oeis.org

1, 2, 8, 0, 72, 0, 0, 338, 3264, 1970, 0, 22964, 0, 0, 323128, 0, 4708320, 0, 10976840, 0, 0, 0, 745778864, 900234724, 0, 2623476242, 0, 0, 110745336312, 178241928596, 0, 0, 7524162792576, 0, 0, 0, 127800022137480, 205691031143924, 0, 0, 0, 0, 0, 40725785296405556
Offset: 0

Views

Author

Paul D. Hanna, Mar 10 2012

Keywords

Comments

Compare g.f. to the Lambert series of A002652: 1 + 2*Sum_{n>=1} Kronecker(n,7)*x^n/(1-x^n).

Examples

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 72*x^4 + 338*x^7 + 3264*x^8 + 1970*x^9 +...
where A(x) = 1 + 1*2*x + 2*4*x^2 + 12*6*x^4 + 169*2*x^7 + 408*8*x^8 + 985*2*x^9 + 5741*4*x^11 + 80782*4*x^14 + 470832*10*x^16 +...+ Pell(n)*A002652(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) - 5*x^3/(1-14*x^3-x^6) + 12*x^4/(1-34*x^4+x^8) - 29*x^5/(1-82*x^5-x^10) - 70*x^6/(1-198*x^6+x^12) + 0*169*13*x^7/(1+478*x^7-x^14) +...).
The values of the symbol Kronecker(n,7) repeat [1,1,-1,1,-1,-1,0, ...].

Links

Crossrefs

Cf. A002652, A205975, A209454, A209456, A204270, A000129 (Pell), A002203.

Programs

  • Mathematica
    terms = 44; s = 1 + 2 Sum[x^n*Fibonacci[n, 2]*KroneckerSymbol[n, 7]/(1 + (-1)^n*x^(2*n) - x^n*(Fibonacci[n - 1, 2] + Fibonacci[n + 1, 2])), {n, 1, terms}] + O[x]^terms; CoefficientList[s, x] (* Jean-François Alcover, Jul 05 2017 *)
A002652[n_]:= If[n < 1, Boole[n == 0], 2*Sum[KroneckerSymbol[-7, d], {d, Divisors[n]}]]; Join[{1}, Table[Fibonacci[n, 2]*A002652[n], {n,1,50}]] (* G. C. Greubel, Jan 03 2017 *)
  • 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 + 2*sum(m=1,n,Pell(m)*kronecker(m,7)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,60,print1(a(n),", "))

Formula

G.f.: 1 + 2*Sum_{n>=1} Pell(n)*Kronecker(n,7)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).
G.f.: 1 + 2*Sum_{n>=1} F(n,2)*Kronecker(n,7)*x^n/(1 + (-1)^n*x^(2*n)-x^n* (F(n-1,2)+F(n+1,2))), where F is the Fibonacci polynomial. - Jean-François Alcover, Jul 05 2017

A205508 a(n) = Pell(n) * A004018(n) for n>=1 with a(0)=1, where A004018(n) is the number of ways of writing n as a sum of 2 squares.

Original entry on oeis.org

1, 4, 8, 0, 48, 232, 0, 0, 1632, 3940, 19024, 0, 0, 267688, 0, 0, 1883328, 9093512, 10976840, 0, 127955424, 0, 0, 0, 0, 15740857452, 25334527696, 0, 0, 356483857192, 0, 0, 2508054264192, 0, 29236023007504, 0, 85200014758320, 411382062287848, 0, 0, 5788584895037376
Offset: 0

Views

Author

Paul D. Hanna, Jan 28 2012

Keywords

Comments

Compare to the g.f. of A004018 given by the Lambert series identity:
1 + 4*Sum_{n>=0} (-1)^n*x^(2*n+1)/(1 - x^(2*n+1)) = (1 + 2*Sum_{n>=1} x^(n^2))^2.

Examples

			 G.f.: A(x) = 1 + 4*x + 8*x^2 + 48*x^4 + 232*x^5 + 1632*x^8 + 3940*x^9 + 19024*x^10 +...
Compare the g.f to the square of the Jacobi theta_3 series:
theta_3(x)^2 = 1 + 4*x + 4*x^2 + 4*x^4 + 8*x^5 + 4*x^8 + 4*x^9 + 8*x^10 +...+ A004018(n)*x^n +...
The g.f. equals the sum:
A(x) = 1 + 4*x/(1-2*x-x^2) - 4*5*x^3/(1-14*x^3-x^6) + 4*29*x^5/(1-82*x^5-x^10) - 4*169*x^7/(1-478*x^7-x^14) + 4*985*x^9/(1-2786*x^9-x^18) - 4*5741*x^11/(1-16238*x^11-x^22) + 4*33461*x^13/(1-94642*x^13-x^26) - 4*195025*x^15/(1-551614*x^15-x^30) +...
which involves odd-indexed Pell and companion Pell numbers.

Crossrefs

Cf. A205507, A204384, A204270, A004018, A000129 (Pell), A002203.

Programs

  • 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+4*sum(m=0,n+1,(-1)^m*Pell(2*m+1)*x^(2*m+1)/(1-A002203(2*m+1)*x^(2*m+1)-x^(4*m+2)+x*O(x^n))))^(1/1),n)}

Formula

G.f.: 1 + 4*Sum_{n>=0} (-1)^n*Pell(2*n+1)*x^(2*n+1) / (1 - A002203(2*n+1)*x^(2*n+1) - x^(4*n+2)), where A002203 is the companion Pell numbers.

A205884 a(n) = Pell(n)*A109064(n) for n >= 1 with a(0)=1.

Original entry on oeis.org

1, -5, 10, 50, -180, -145, -700, 5070, 10200, -34475, 11890, -344460, 415800, 2007660, -2423460, 1950250, -25895760, 90935120, 96047350, -662510900, -239916420, -2316837900, 5593341480, 24756454910, -27166986000, -6558690605, -190008957720, 764537004500
Offset: 0

Views

Author

Paul D. Hanna, Feb 01 2012

Keywords

Comments

Compare to g.f. of A109064, which is the Lambert series identity:
1 - 5*Sum_{n>=1} L(n,5)*n*x^n/(1-x^n) = eta(x)^5/eta(x^5).
Here L(n,5) is the Legendre symbol given by A080891(n).

Examples

			G.f.: A(x) = 1 - 5*x + 10*x^2 + 50*x^3 - 180*x^4 - 145*x^5 - 700*x^6 + ...
where A(x) = 1 - 1*5*x + 2*5*x^2 + 5*10*x^3 - 12*15*x^4 - 29*5*x^5 - 70*10*x^6 + 169*30*x^7 + 408*25*x^8 + ... + Pell(n)*A109064(n)*x^n + ...
The g.f. is illustrated by:
A(x) = 1 - 5*(+1)*1*1*x/(1-2*x-x^2) - 5*(-1)*2*2*x^2/(1-6*x^2+x^4) - 5*(-1)*3*5*x^3/(1-14*x^3-x^6) - 5*(+1)*4*12*x^4/(1-34*x^4+x^8) - 5*(0)*5*29*x^5/(1-82*x^5-x^10) - 5*(+1)*6*70*x^6/(1-198*x^6+x^12) + ...
The values of the Legendre symbol L(n,5) repeat: [1,-1,-1,1,0, ...].
The companion Pell numbers (A002203) begin: [2,6,14,34,82,198,478,1154,2786,6726,16238,39202,94642,...].

Links

Crossrefs

Cf. A109064, A080891, A000129 (Pell), A002203 (companion Pell), A205882 (variant), A204270.

Programs

  • Mathematica
    pell[n_] := ((1+Sqrt[2])^n - (1-Sqrt[2])^n)/(2*Sqrt[2]) // Simplify; (* b = A109064 *); b[0] = 1; b[n_] := b[n] = Sum[DivisorSum[j, #*If[Divisible[#, 5], -4, -5] &]*b[n - j], {j, 1, n}]/n; a[0] = 1; a[n_] := pell[n]*b[n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 24 2017 *)
  • PARI
    {A109064(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^5/eta(x^5+A), n))}
{a(n)=if(n==0,1,Pell(n)*A109064(n))}
  • PARI
    {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)}
{a(n)=polcoeff(1-5*sum(m=1, n, kronecker(m, 5)*m*Pell(m)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))), n)}

Formula

G.f.: 1 - 5*Sum_{n>=1} Pell(n)*L(n,5)*n*x^n / (1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where L(n,5) is the Legendre symbol, Pell(n) = A000129(n), and A002203 is the companion Pell numbers.

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.

Original entry on oeis.org

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
Offset: 0

Views

Author

Paul D. Hanna, Mar 09 2012

Keywords

Comments

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

Examples

			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) +...).

Links

Crossrefs

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

Programs

  • 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 *)
  • 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),", "))

Formula

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

A209445 a(n) = Pell(n)*A001227(n) for n >= 1, where A001227(n) is the number of odd divisors of n.

Original entry on oeis.org

1, 2, 10, 12, 58, 140, 338, 408, 2955, 4756, 11482, 27720, 66922, 161564, 780100, 470832, 2273378, 8232630, 13250218, 31988856, 154455860, 186444716, 450117362, 1086679440, 3935214363, 6333631924, 30581480180, 36915112104, 89120964298, 430314081400, 519435045698
Offset: 1

Views

Author

Paul D. Hanna, Mar 09 2012

Keywords

Comments

Compare g.f. to the Lambert series of A001227: Sum_{n>=1} x^(2*n-1)/(1 - x^(2*n-1)).

Examples

			G.f.: A(x) = x + 2*x^2 + 10*x^3 + 12*x^4 + 58*x^5 + 140*x^6 + 338*x^7 + ...
where A(x) = 1*1*x + 2*1*x^2 + 5*2*x^3 + 12*1*x^4 + 29*2*x^5 + 70*2*x^6 + 169*2*x^7 + 408*1*x^8 + ... + Pell(n)*A001227(n)*x^n + ...
The g.f. is also given by the identity:
A(x) = 1*x/(1-2*x-x^2) + 5*x^3/(1-14*x^3-x^6) + 29*x^5/(1-82*x^5-x^10) + 169*x^7/(1-478*x^7-x^14) + 985*x^9/(1-2786*x^9-x^18) + 5741*x^11/(1-16238*x^11-x^22) + ...
which involves odd-indexed Pell and A002203 numbers.

Links

Crossrefs

Cf. A001227, A205965, A209444, A209446, A204270.

Programs

  • Mathematica
    A001227[n_]:= Sum[Mod[d, 2], {d, Divisors[n]}]; Table[Fibonacci[n, 2]*A001227[n], {n, 1, 1000}] (* G. C. Greubel, Jan 02 2018 *)
  • 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(sum(m=1,n,Pell(2*m-1)*x^(2*m-1)/(1-A002203(2*m-1)*x^(2*m-1)-x^(4*m-2)+x*O(x^n))),n)}
for(n=1,40,print1(a(n),", "))

Formula

G.f.: Sum_{n>=1} Pell(2*n-1)*x^(2*n-1)/(1 - A002203(2*n-1)*x^(2*n-1)-x^(4*n-2)), where A002203(n) = Pell(n-1) + Pell(n+1).
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