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).

A205966 a(n) = Fibonacci(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.

Original entry on oeis.org

1, 6, 0, 12, 18, 0, 0, 156, 0, 204, 0, 0, 864, 2796, 0, 0, 5922, 0, 0, 50172, 0, 131352, 0, 0, 0, 450150, 0, 1178508, 3813732, 0, 0, 16155228, 0, 0, 0, 0, 89582112, 289893804, 0, 758951832, 0, 0, 0, 5201933244, 0, 0, 0, 0, 28845161856, 140017356882, 0, 0
Offset: 0

Views

Author

Paul D. Hanna, Feb 03 2012

Keywords

Comments

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

Examples

			G.f.: A(x) = 1 + 6*x + 12*x^3 + 18*x^4 + 156*x^7 + 204*x^9 + 864*x^12 +...
where A(x) = 1 + 1*6*x + 2*6*x^3 + 3*6*x^4 + 13*12*x^7 + 34*6*x^9 + 144*6*x^12 +...+ Fibonacci(n)*A004016(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 6*( 1*x/(1-x-x^2) - 1*x^2/(1-3*x^2+x^4) + 3*x^4/(1-7*x^4+x^8) - 5*x^5/(1-11*x^5-x^10) + 13*x^7/(1-29*x^7-x^14) - 21*x^8/(1-47*x^8-x^16) +...).
The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].

Links

Crossrefs

Cf. A004016, A205965, A205967, A203847, A000204 (Lucas).
Cf. A209446 (Pell variant).

Programs

  • Mathematica
    A004016[n_] := SeriesCoefficient[(QPochhammer[q]^3 + 9 q QPochhammer[q^9]^3)/QPochhammer[q^3], {q, 0, n}]; Join[{1}, Table[Fibonacci[n]*b[n], {n,1,50}]] (* G. C. Greubel, Mar 05 2017 *)
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 + 6*sum(m=1,n,kronecker(m,3)*fibonacci(m)*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 + 6*Sum_{n>=1} Fibonacci(n)*Kronecker(n,3)*x^n/(1 - Lucas(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).
Showing 1-3 of 3 results.

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