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).

A205964 a(n) = Fibonacci(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, 112, 896, 3408, 10080, 25088, 71552, 195888, 411808, 776160, 1896768, 4580352, 8194144, 14525056, 34433280, 73890768, 125562528, 219081856, 458906560, 968315040, 1686909952, 2642197824, 5579174016, 12110579712, 18907500400, 29884043168, 64236542720
Offset: 0

Views

Author

Paul D. Hanna, Feb 03 2012

Keywords

Comments

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

Examples

			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 + 1*112*x^2 + 2*448*x^3 + 3*1136*x^4 + 5*2016*x^5 + 8*3136*x^6 + 13*5504*x^7 + 21*9328*x^8 +...+ Fibonacci(n)*A000143(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 16*( 1*1*x/(1+x-x^2) + 1*8*x^2/(1-3*x^2+x^4) + 2*27*x^3/(1+4*x^3-x^6) + 3*64*x^4/(1-7*x^4+x^8) + 5*125*x^5/(1+11*x^5-x^10) + 8*216*x^6/(1-18*x^6+x^12) + 13*343*x^7/(1+29*x^7-x^14) +...).

Links

Crossrefs

Cf. A000143, A205507, A205963, A203847, A000204 (Lucas).
Cf. A209444 (Pell variant).

Programs

  • Mathematica
    Join[{1}, Table[Fibonacci[n]*SquaresR[8, n], {n,1,30}]] (* G. C. Greubel, Mar 05 2017 *)
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1+16*sum(m=1,n,fibonacci(m)*m^3*x^m/(1-Lucas(m)*(-x)^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
for(n=0,31,print1(a(n),", "))

Formula

G.f.: 1 + 16*Sum_{n>=1} Fibonacci(n)*n^3*x^n/(1 - Lucas(n)*(-x)^n + (-1)^n*x^(2*n)).

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)).
Showing 1-3 of 3 results.

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