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.

Showing 1-3 of 3 results.

A205974 a(n) = Fibonacci(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, 6, 0, 0, 26, 84, 68, 0, 356, 0, 0, 0, 0, 5922, 0, 0, 0, 0, 0, 0, 114628, 0, 150050, 0, 0, 635622, 2056916, 0, 0, 17426472, 0, 0, 0, 29860704, 96631268, 0, 0, 0, 0, 0, 1733977748, 2805634932, 0, 0, 0, 0, 15557484098, 0, 0, 0, 213265164692, 0, 0
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2012

Keywords

Comments

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

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^4 + 26*x^7 + 84*x^8 + 68*x^9 + 356*x^11 +...
where A(x) = 1 + 1*2*x + 3*2*x^4 + 13*2*x^7 + 21*4*x^8 + 34*2*x^9 + 89*4*x^11 + 987*6*x^16 + 28657*4*x^23 +...+ Fibonacci(n)*A033719(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 2*( 1*x/(1+x-x^2) + 1*x^2/(1-3*x^2+x^4) - 2*x^3/(1+4*x^3-x^6) + 3*x^4/(1-7*x^4+x^8) - 5*x^5/(1+11*x^5-x^10) - 8*x^6/(1-18*x^6+x^12) + 0*13*x^7/(1+29*x^7-x^14) +...).
The values of the symbol Kronecker(n,7) repeat [1,1,-1,1,-1,-1,0, ...].

Crossrefs

Cf. A033719, A205973, A205975, A203847, A000204 (Lucas).
Cf. A209454 (Pell variant).

Programs

  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 + 2*sum(m=1,n,fibonacci(m)*kronecker(m,7)*x^m/(1-Lucas(m)*(-x)^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,40,print1(a(n),", "))

Formula

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

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).

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
Showing 1-3 of 3 results.

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