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

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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).

A205971 a(n) = Fibonacci(n)*A034896(n) for n >= 1, with a(0)=1, where A034896 lists the number of solutions to a^2 + b^2 + 3*c^2 + 3*d^2 = n.

Original entry on oeis.org

1, 4, 4, 8, 60, 120, 32, 416, 1092, 136, 1320, 4272, 2880, 13048, 12064, 14640, 114492, 114984, 10336, 334480, 811800, 350272, 850128, 2751072, 2411136, 9303100, 6798008, 785672, 50849760, 61707480, 19968960, 172322432, 531507396, 169179744, 410607864
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2012

Keywords

Comments

Compare g.f. to the Lambert series of A034896:
1 + 4*Sum_{n>=1} Chi(n,3)*n*x^n/(1 - (-x)^n).
Here Chi(n,3) = principal Dirichlet character of n modulo 3.

Examples

			G.f.: A(x) = 1 + 4*x + 4*x^2 + 8*x^3 + 60*x^4 + 120*x^5 + 32*x^6 + ...
where A(x) = 1 + 1*4*x + 1*4*x^2 + 2*4*x^3 + 3*20*x^4 + 5*24*x^5 + 8*4*x^6 + ... + Fibonacci(n)*A034896(n)*x^n + ...
The g.f. is also given by the identity:
A(x) = 1 + 4*( 1*1*x/(1+x-x^2) + 1*2*x^2/(1-3*x^2+x^4) + 3*4*x^4/(1-7*x^4+x^8) + 5*5*x^5/(1+11*x^5-x^10) + 13*7*x^7/(1+29*x^7-x^14) + 21*8*x^8/(1-47*x^8+x^16) + ...).
The values of the Dirichlet character Chi(n,3) repeat [1,1,0,...].

Links

Crossrefs

Cf. A034896, A205882, A205967, A205970, A205972, A203847, A000204 (Lucas).
Cf. A209451 (Pell variant).

Programs

  • Mathematica
    A034896[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^3])^2, {q, 0, n}]; Join[{1}, Table[Fibonacci[n]*A034896[n], {n, 1, 50}]] (* G. C. Greubel, Dec 24 2017 *)
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 + 4*sum(m=1,n,fibonacci(m)*kronecker(m,3)^2*m*x^m/(1-Lucas(m)*(-x)^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,61,print1(a(n),", "))

Formula

G.f.: 1 + 4*Sum_{n>=1} Fibonacci(n)*Chi(n,3)*n*x^n/(1 - Lucas(n)*(-x)^n + (-1)^n*x^(2*n)).

A205973 a(n) = Fibonacci(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, 27, -18, -351, 1080, 216, -5850, 9639, -306, -35640, 96120, -16848, -356490, 508950, 131760, -1821015, 4139424, 69768, -13621698, 18996120, -4925700, -57383640, 136178064, 21282912, -405810225, 557193870, -1767762, -1859194350, 3887571240, -539161920
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 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 + 27*x^2 - 18*x^3 - 351*x^4 + 1080*x^5 + 216*x^6 +...
where A(x) = 1 - 1*9*x + 1*27*x^2 - 2*9*x^3 - 3*117*x^4 + 5*216*x^5 + 8*27*x^6 - 13*450*x^7 + 21*459*x^8 +...+ Fibonacci(n)*A109041(n)*^n +...
The g.f. is also given by the identity:
A(x) = 1 - 9*( 1*1*x/(1-x-x^2) - 1*4*x^2/(1-3*x^2+x^4) + 3*16*x^4/(1-7*x^4+x^8) - 5*25*x^5/(1-11*x^5-x^10) + 13*49*x^7/(1-29*x^7-x^14) - 21*64*x^8/(1-47*x^8+x^16) +...).
The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].

Crossrefs

Cf. A109041, A205972, A205974, A203847, A000204 (Lucas).
Cf. A209453 (Pell variant).

Programs

  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 - 9*sum(m=1,n,fibonacci(m)*kronecker(m,3)*m^2*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 - 9*Sum_{n>=1} Fibonacci(n)*Kronecker(n,3)*n^2*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).

A209452 a(n) = Pell(n)*A122859(n) for n>=1, with a(0)=1, where A122859 lists the coefficients in phi(-q)^3/phi(-q^3) and phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -6, 24, -30, -72, 0, 840, -2028, 4896, -5910, 0, 0, -83160, -401532, 1938768, 0, -2824992, 0, 32930520, -79501308, 0, -463367580, 0, 0, 6520076640, -7870428726, 76003583088, -45872220270, -221490672624, 0, 0, -3116610274188, 7524162792576, 0, 0, 0, -127800022137480
Offset: 0

Views

Author

Paul D. Hanna, Mar 10 2012

Keywords

Comments

Compare the g.f. to the Lambert series of A122859: 1 - 6*Sum_{n>=1} Kronecker(n,3)*x^n/(1+x^n).

Examples

			G.f.: A(x) = 1 - 6*x + 24*x^2 - 30*x^3 - 72*x^4 + 840*x^6 - 2028*x^7 + ...
where A(x) = 1 - 1*6*x + 2*12*x^2 - 5*6*x^3 - 12*6*x^4 + 70*12*x^6 - 169*12*x^7 + 408*12*x^8 - 985*6*x^9 + ... + Pell(n)*A122859(n)*x^n + ...
The g.f. is also given by the identity:
A(x) = 1 - 6*( 1*x/(1+2*x-x^2) - 2*x^2/(1+6*x^2+x^4) + 12*x^4/(1+34*x^4+x^8) - 29*x^5/(1+82*x^5-x^10) + 169*x^7/(1+478*x^7-x^14) - 408*x^8/(1+1154*x^8+x^16) + ...).
The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].

Links

Crossrefs

Cf. A122859, A205972, A209451, A209453, A209446, A209449, A204270, A000129 (Pell), A002203.

Programs

  • Mathematica
    A122859[n_]:= SeriesCoefficient[EllipticTheta[4, 0, q]^3/EllipticTheta[4, 0, q^3], {q, 0, n}]; Join[{1}, Table[Fibonacci[n, 2]*A122859[n], {n, 1, 50}]] (* G. C. Greubel, Jan 02 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 - 6*sum(m=1,n,Pell(m)*kronecker(m,3)*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 - 6*Sum_{n>=1} Pell(n)*Kronecker(n,3)*x^n/(1 + A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).
Showing 1-4 of 4 results.

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