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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A203847 a(n) = tau(n)*Fibonacci(n), where tau(n) = A000005(n), the number of divisors of n.

Original entry on oeis.org

1, 2, 4, 9, 10, 32, 26, 84, 102, 220, 178, 864, 466, 1508, 2440, 4935, 3194, 15504, 8362, 40590, 43784, 70844, 57314, 370944, 225075, 485572, 785672, 1906866, 1028458, 6656320, 2692538, 13069854, 14098312, 22811548, 36909860, 134373168, 48315634, 156352676, 252983944
Offset: 1

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Author

Paul D. Hanna, Jan 11 2012

Keywords

Comments

Compare g.f. to the Lambert series identity: Sum_{n>=1} x^n/(1-x^n) = Sum_{n>=1} tau(n)*x^n.
Related identities:
(1) Sum_{n>=1} n^k*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_{k}(n)*Fibonacci(n)*x^n for k>=0.
(2) Sum_{n>=1} phi(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} n*Fibonacci(n)*x^n.
(3) Sum_{n>=1} moebius(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = x.
(4) Sum_{n>=1} lambda(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} Fibonacci(n^2)*x^(n^2).

Examples

			G.f.: A(x) = x + 2*x^2 + 4*x^3 + 9*x^4 + 10*x^5 + 32*x^6 + 26*x^7 +...
where A(x) = x/(1-x-x^2) + 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) +...+ Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...

Links

Crossrefs

Cf. A203848, A203849, A203838, A204060, A000005 (tau), A000204 (Lucas), A000045.
Cf. A205507, A205882, A205963, A205964, A205965, A205966.
Cf. A205967, A205968, A205969, A205970, A205971.
Cf. A205972, A205973, A205974, A205975, A205976.
Cf. A204291, A203801, A203850, A203860, A203861.

Programs

  • Mathematica
    Table[DivisorSigma[0, n]*Fibonacci[n], {n, 50}] (* G. C. Greubel, Jul 17 2018 *)
  • PARI
    {a(n)=sigma(n,0)*fibonacci(n)}
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1,n,fibonacci(m)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
  • PARI
    a(n) = numdiv(n)*fibonacci(n); \\ Michel Marcus, Jul 18 2018

Formula

G.f.: Sum_{n>=1} Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} tau(n)*Fibonacci(n)*x^n, where Lucas(n) = A000204(n).

A205975 a(n) = Fibonacci(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, 4, 0, 18, 0, 0, 26, 168, 68, 0, 356, 0, 0, 1508, 0, 9870, 0, 10336, 0, 0, 0, 141688, 114628, 0, 150050, 0, 0, 1906866, 2056916, 0, 0, 26139708, 0, 0, 0, 89582112, 96631268, 0, 0, 0, 0, 0, 1733977748, 8416904796, 0, 14690495224, 0, 0, 15557484098
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2012

Keywords

Comments

Compare the 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 + 4*x^2 + 18*x^4 + 26*x^7 + 168*x^8 + 68*x^9 + 356*x^11 +...
where A(x) = 1 + 1*2*x + 1*4*x^2 + 3*6*x^4 + 14*2*x^7 + 21*8*x^8 + 34*2*x^9 + 89*4*x^11 + 377*4*x^14 + 987*10*x^16 +...+ Fibonacci(n)*A002652(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. A002652, A205974, A205976, A203847, A000204 (Lucas).
Cf. A209455 (Pell variant).

Programs

  • Mathematica
    terms = 50; s = 1 + 2 Sum[Fibonacci[n]*KroneckerSymbol[n, 7]*x^n/(1 - LucasL[n]*x^n + (-1)^n*x^(2*n)), {n, 1, terms}] + O[x]^terms; CoefficientList[s, x] (* Jean-François Alcover, Jul 05 2017 *)
  • 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=0,60,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)).

A209456 a(n) = Pell(n)*A028594(n) for n>=1, with a(0)=1, where A028594 lists the coefficients in (theta_3(x)*theta_3(7*x)+theta_2(x)*theta_2(7*x))^2.

Original entry on oeis.org

1, 4, 24, 80, 336, 696, 3360, 676, 24480, 51220, 171216, 275568, 1552320, 1873816, 969384, 18722400, 58383168, 81841608, 428096760, 530008720, 2687063904, 617823440, 13424019552, 21605633376, 130401532800, 162655527004, 532025081616, 1223259207200
Offset: 0

Views

Author

Paul D. Hanna, Mar 10 2012

Keywords

Comments

Compare g.f. to the Lambert series of A028594:
1 + 4*Sum_{n>=1} Chi(n,7)*n*x^n/(1-x^n).
Here Chi(n,7) = principal Dirichlet character of n modulo 7.

Examples

			G.f.: A(x) = 1 + 4*x + 24*x^2 + 80*x^3 + 336*x^4 + 696*x^5 + 3360*x^6 +...
where A(x) = 1 + 1*4*x + 2*12*x^2 + 5*16*x^3 + 12*28*x^4 + 29*24*x^5 + 70*48*x^6 + 169*4*x^7 + 408*60*x^8 + 985*52*x^9 +...+ Pell(n)*A028594(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) + 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) + 0*169*7*x^7/(1+478*x^7-x^14) +...).
The values of the Dirichlet character Chi(n,7) repeat [1,1,1,1,1,1,0, ...].

Links

Crossrefs

Cf. A028594, A205976, A209455, A204270, A000129 (Pell), A002203.

Programs

  • Mathematica
    A028594[n_]:= If[n < 1, Boole[n == 0], 4*Sum[If[Mod[d, 7] > 0, d, 0], {d, Divisors@n}]]; Join[{1}, Table[Fibonacci[n, 2]*A028594[n], {n,1,50}]] (* G. C. Greubel, Jan 03 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 + 4*sum(m=1,n,Pell(m)*kronecker(m,7)^2*m*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,35,print1(a(n),", "))

Formula

G.f.: 1 + 4*Sum_{n>=1} Pell(n)*Chi(n,7)*n*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).
Showing 1-3 of 3 results.

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