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 1-4 of 4 results.

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

Views

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).

A203848 a(n) = sigma(n)*Fibonacci(n), where sigma(n) = A000203(n), the sum of divisors of n.

Original entry on oeis.org

1, 3, 8, 21, 30, 96, 104, 315, 442, 990, 1068, 4032, 3262, 9048, 14640, 30597, 28746, 100776, 83620, 284130, 350272, 637596, 687768, 2782080, 2325775, 5098506, 7856720, 17797416, 15426870, 59906880, 43080608, 137233467, 169179744, 307955898, 442918320, 1358662032
Offset: 1

Views

Author

Paul D. Hanna, Jan 12 2012

Keywords

Comments

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

Examples

			G.f.: A(x) = x + 3*x^2 + 8*x^3 + 21*x^4 + 30*x^5 + 96*x^6 + 104*x^7 +...
where A(x) = x/(1-x-x^2) + 2*1*x^2/(1-3*x^2+x^4) + 3*2*x^3/(1-4*x^3-x^6) + 4*3*x^4/(1-7*x^4+x^8) + 5*5*x^5/(1-11*x^5-x^10) + 6*8*x^6/(1-18*x^6+x^12) +...+ n*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...

Links

Crossrefs

Cf. A203847, A203849, A203838, A000203 (sigma), A000204 (Lucas), A000045.

Programs

  • Magma
    [DivisorSigma(1, n)*Fibonacci(n): n in [1..40]]; // Vincenzo Librandi, Aug 12 2016
  • Mathematica
    Table[DivisorSigma[1, n] Fibonacci[n], {n, 40}] (* Wesley Ivan Hurt, Aug 10 2016 *)
  • PARI
    {a(n)=sigma(n)*fibonacci(n)}
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1,n,m*fibonacci(m)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}

Formula

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

A203849 a(n) = sigma_2(n)*Fibonacci(n), where sigma_2(n) = A001157(n), the sum of squares of divisors of n.

Original entry on oeis.org

1, 5, 20, 63, 130, 400, 650, 1785, 3094, 7150, 10858, 30240, 39610, 94250, 158600, 336567, 463130, 1175720, 1513522, 3693690, 5473000, 10803710, 15188210, 39412800, 48841275, 103184050, 161062760, 333701550, 432980818, 1081652000, 1295110778, 2973391785, 4299985160
Offset: 1

Views

Author

Paul D. Hanna, Jan 12 2012

Keywords

Comments

Compare g.f. to the Lambert series identity: Sum_{n>=1} n^2*x^n/(1-x^n) = Sum_{n>=1} sigma_2(n)*x^n.

Examples

			G.f.: A(x) = x + 5*x^2 + 20*x^3 + 63*x^4 + 130*x^5 + 400*x^6 + 650*x^7 +...
where A(x) = x/(1-x-x^2) + 2^2*1*x^2/(1-3*x^2+x^4) + 3^2*2*x^3/(1-4*x^3-x^6) + 4^2*3*x^4/(1-7*x^4+x^8) + 5^2*5*x^5/(1-11*x^5-x^10) + 6^2*8*x^6/(1-18*x^6+x^12) +...+ n^2*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...

Links

Crossrefs

Cf. A203847, A203848, A203838, A001157 (sigma_2), A000204 (Lucas), A000045.

Programs

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

Formula

G.f.: Sum_{n>=1} n^2*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_2(n)*fibonacci(n)*x^n, where Lucas(n) = A000204(n).

A204273 a(n) = sigma_3(n)*Pell(n), where sigma_3(n) = A001158(n), the sum of cubes of divisors of n.

Original entry on oeis.org

1, 18, 140, 876, 3654, 17640, 58136, 238680, 745645, 2696652, 7647012, 28329840, 73547278, 250101072, 688048200, 2203964592, 5585689746, 18696302730, 45448247740, 147116748744, 371929710880, 1117549627704, 2738514030408, 8899904613600
Offset: 1

Views

Author

Paul D. Hanna, Jan 14 2012

Keywords

Comments

Compare g.f. to the Lambert series identity: Sum_{n>=1} n^3*x^n/(1-x^n) = Sum_{n>=1} sigma_3(n)*x^n.

Examples

			G.f.: A(x) = x + 18*x^2 + 140*x^3 + 876*x^4 + 3654*x^5 + 17640*x^6 + ...
where A(x) = x/(1-2*x-x^2) + 2^3*2*x^2/(1-6*x^2+x^4) + 3^3*5*x^3/(1-14*x^3-x^6) + 4^3*12*x^4/(1-34*x^4+x^8) + 5^3*29*x^5/(1-82*x^5-x^10) + 6^3*70*x^6/(1-198*x^6+x^12) + ... + n^3*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) + ...

Links

Crossrefs

Cf. A203838, A204270, A204271, A204272, A204274, A204275, A001158 (sigma_3), A002203, A000045.

Programs

  • Mathematica
    Table[DivisorSigma[3, n] Fibonacci[n, 2], {n, 1, 50}] (* G. C. Greubel, Jan 05 2018 *)
  • PARI
    /* Subroutines used in PARI programs below: */
{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)}
  • PARI
    {a(n)=sigma(n,3)*Pell(n)}
  • PARI
    {a(n)=polcoeff(sum(m=1,n,m^3*Pell(m)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}

Formula

G.f.: Sum_{n>=1} n^3*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_3(n)*Pell(n)*x^n, where Pell(n) = A000129(n) and A002203 is the companion Pell numbers.
Showing 1-4 of 4 results.

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