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 1-5 of 5 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).

A205968 a(n) = Fibonacci(n)*A008655(n) for n >= 1, with a(0)=1, where A008655 lists the coefficients in (theta_3(x)*theta_3(3*x) + theta_2(x)*theta_2(3*x))^4.

Original entry on oeis.org

1, 24, 216, 1776, 5256, 15120, 63936, 107328, 294840, 823344, 1496880, 2845152, 9334656, 12291216, 28012608, 68251680, 110883528, 188343792, 563167296, 688359840, 1493387280, 3343696512, 5095667232, 8368761024, 24087248640, 28361250600, 57633511824, 128471514096
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2012

Keywords

Comments

Compare g.f. to the Lambert series of A008655:
1 + Sum_{n>=1} 24*n^3*x^n/(1-x^n) + 8*(3*n)^3*x^(3*n)/(1-x^(3*n)).

Examples

			G.f.: A(x) = 1 + 24*x + 216*x^2 + 1776*x^3 + 5256*x^4 + 15120*x^5 + ...
where A(x) = 1 + 1*24*x + 1*216*x^2 + 2*888*x^3 + 3*1752*x^4 + 5*3024*x^5 + ... + Fibonacci(n)*A008655(n)*x^n + ...

Links

Crossrefs

Cf. A008655, A205967, A205969, A203847, A000204 (Lucas).
Cf. A209448 (Pell variant).

Programs

  • Mathematica
    A008655 := CoefficientList[Series[((EllipticTheta[3, 0, q]^3 + EllipticTheta[3, Pi/3, q]^3 + EllipticTheta[3, 2 Pi/3, q]^3)^4/(3* EllipticTheta[3, 0, q^3])^4), {q, 0, 250}], q]; b := Table[A008655[[n]], {n, 1, 120}][[1 ;; ;; 2]]; Join[{1}, Table[Fibonacci[n]*b[[n + 1]], {n, 1, 50}]] (* G. C. Greubel, Jul 16 2018 *)
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 + sum(m=1,n, 24*fibonacci(m)*m^3*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n)) + 8*fibonacci(3*m)*(3*m)^3*x^(3*m)/(1-Lucas(3*m)*x^(3*m)+(-1)^m*x^(6*m) +x*O(x^n)) ),n)}
for(n=0,40,print1(a(n),", "))

Formula

G.f.: 1 + Sum_{n>=1} 24*Fibonacci(n)*n^3*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) + 8*Fibonacci(3*n)*(3*n)^3*x^(3*n)/(1 - Lucas(3*n)*x^(3*n) + (-1)^n*x^(6*n)).

A205970 a(n) = Fibonacci(n)*A132973(n) for n>=1, with a(0)=1, where A132973 lists the coefficients in psi(-q)^3/psi(-q^3) and where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -3, 3, -6, 9, 0, 24, -78, 63, -102, 0, 0, 432, -1398, 2262, 0, 2961, 0, 7752, -25086, 0, -65676, 0, 0, 139104, -225075, 728358, -589254, 1906866, 0, 0, -8077614, 6534927, 0, 0, 0, 44791056, -144946902, 234529014, -379475916, 0, 0, 1607485776, -2600966622, 0
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2012

Keywords

Comments

Compare g.f. to the Lambert series of A132973:
1 - 3*Sum_{n>=0} x^(6*n+1)/(1+x^(6*n+1)) - x^(6*n+5)/(1+x^(6*n+5)).

Examples

			G.f.: A(x) = 1 - 3*x + 3*x^2 - 6*x^3 + 9*x^4 + 24*x^6 - 78*x^7 + 63*x^8 +...
where A(x) = 1 - 1*3*x + 1*3*x^2 - 2*3*x^3 + 3*3*x^4 + 8*3*x^6 - 13*6*x^7 + 21*3*x^8 +...+ Fibonacci(n)*A132973(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 - 3*( 1*x/(1+x-x^2) - 5*x^5/(1+11*x^5-x^10) + 13*x^7/(1+29*x^7-x^14) - 89*x^11/(1+199*x^11-x^22) + 233*x^13/(1+521*x^13-x^26) - 1597*x^17/(1+3571*x^17-x^34) +...).

Links

Crossrefs

Cf. A132973, A205969, A205971, A203847.
Cf. A209450 (Pell variant).

Programs

  • Mathematica
    A132973:= CoefficientList[Series[(-1)^(-1/4)*EllipticTheta[2, 0, I*Sqrt[q]]^3/EllipticTheta[2, 0, I*Sqrt[q^3]]/4, {q, 0, 60}], q]; Table[If[n == 0, 1, Fibonacci[n]*A132973[[n + 1]]], {n, 0, 50}] (* G. C. Greubel, Dec 03 2017 *)
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 - 3*sum(m=0,n, fibonacci(6*m+1)*x^(6*m+1)/(1+Lucas(6*m+1)*x^(6*m+1)-x^(12*m+2) +x*O(x^n)) - fibonacci(6*m+5)*x^(6*m+5)/(1+Lucas(6*m+5)*x^(6*m+5)-x^(12*m+10) +x*O(x^n)) ),n)}
for(n=0,61,print1(a(n),", "))

Formula

G.f.: 1 - 3*Sum_{n>=0} Fibonacci(6*n+1)*x^(6*n+1)/(1 + Lucas(6*n+1) * x^(6*n+1) - x^(12*n+2)) - Fibonacci(6*n+5)*x^(6*n +5)/(1 + Lucas(6*n+5) * x^(6*n+5) - x^(12*n+10)).

A205972 a(n) = Fibonacci(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, 12, -12, -18, 0, 96, -156, 252, -204, 0, 0, -864, -2796, 9048, 0, -5922, 0, 31008, -50172, 0, -131352, 0, 0, 556416, -450150, 2913432, -1178508, -3813732, 0, 0, -16155228, 26139708, 0, 0, 0, -89582112, -289893804, 938116056, -758951832, 0, 0, 6429943104
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 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 + 12*x^2 - 12*x^3 - 18*x^4 + 96*x^6 - 156*x^7 +...
where A(x) = 1 - 1*6*x + 1*12*x^2 - 2*6*x^3 - 3*6*x^4 + 8*12*x^6 - 13*12*x^7 + 21*12*x^8 - 34*6*x^9 +...+ Fibonacci(n)*A122859(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. A122859, A205971, A205973, A205966, A205969, A203847, A000204 (Lucas).
Cf. A209452 (Pell variant).

Programs

  • Mathematica
    A122859:= CoefficientList[Series[EllipticTheta[3, 0, -q]^3/EllipticTheta[3, 0, -q^3], {q, 0, 60}], q]; Table[If[n == 1, 1, Fibonacci[n - 1]*A122859[[n]]], {n, 1, 50}] (* G. C. Greubel, Dec 03 2017 *)
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 - 6*sum(m=1,n,fibonacci(m)*kronecker(m,3)*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 - 6*Sum_{n>=1} Fibonacci(n)*Kronecker(n,3)*x^n/(1 + Lucas(n)*x^n + (-1)^n*x^(2*n)).

A209449 a(n) = Pell(n)*A113973(n) for n>=1, with a(0)=1, where A113973 lists the coefficients in phi(x^3)^3/phi(x) and phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 8, -10, 24, 0, 280, -676, 1632, -1970, 0, 0, 27720, -133844, 646256, 0, 941664, 0, 10976840, -26500436, 0, -154455860, 0, 0, 2173358880, -2623476242, 25334527696, -15290740090, 73830224208, 0, 0, -1038870091396, 2508054264192, 0, 0, 0, 42600007379160
Offset: 0

Views

Author

Paul D. Hanna, Mar 10 2012

Keywords

Comments

Compare g.f. to the Lambert series of A113973:
1 - 2*Sum_{n>=1} Kronecker(n,3)*x^n/(1 - (-x)^n).

Examples

			G.f.: A(x) = 1 - 2*x + 8*x^2 - 10*x^3 + 24*x^4 + 280*x^6 - 676*x^7 +...
where A(x) = 1 - 1*2*x + 2*4*x^2 - 5*2*x^3 + 12*2*x^4 + 70*4*x^6 - 169*4*x^7 + 408*4*x^8 +...+ Pell(n)*A113973(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 - 2*( 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. A113973, A205969, A209446, A209448, A209450, A204270, A000129 (Pell), A002203.

Programs

  • Mathematica
    A113973:= CoefficientList[Series[EllipticTheta[3, 0, q^3]^3 /EllipticTheta[3, 0, q], {q, 0, 60}], q]; Table[If[n == 0, 1, Fibonacci[n, 2]*A113973[[n + 1]]], {n, 0, 50}] (* G. C. Greubel, Dec 03 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 - 2*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=0,60,print1(a(n),", "))

Formula

G.f.: 1 - 2*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 1-5 of 5 results.

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