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).

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)).

A205974 a(n) = Fibonacci(n)*A033719(n) for n>=1, with a(0)=1, where A033719 lists the coefficients in theta_3(q)*theta_3(q^7).

Original entry on oeis.org

1, 2, 0, 0, 6, 0, 0, 26, 84, 68, 0, 356, 0, 0, 0, 0, 5922, 0, 0, 0, 0, 0, 0, 114628, 0, 150050, 0, 0, 635622, 2056916, 0, 0, 17426472, 0, 0, 0, 29860704, 96631268, 0, 0, 0, 0, 0, 1733977748, 2805634932, 0, 0, 0, 0, 15557484098, 0, 0, 0, 213265164692, 0, 0
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2012

Keywords

Comments

Compare g.f. to the Lambert series of A033719:
1 + 2*Sum_{n>=1} Kronecker(n,7)*x^n/(1-(-x)^n).

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^4 + 26*x^7 + 84*x^8 + 68*x^9 + 356*x^11 +...
where A(x) = 1 + 1*2*x + 3*2*x^4 + 13*2*x^7 + 21*4*x^8 + 34*2*x^9 + 89*4*x^11 + 987*6*x^16 + 28657*4*x^23 +...+ Fibonacci(n)*A033719(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 2*( 1*x/(1+x-x^2) + 1*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) + 0*13*x^7/(1+29*x^7-x^14) +...).
The values of the symbol Kronecker(n,7) repeat [1,1,-1,1,-1,-1,0, ...].

Crossrefs

Cf. A033719, A205973, A205975, A203847, A000204 (Lucas).
Cf. A209454 (Pell variant).

Programs

  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 + 2*sum(m=1,n,fibonacci(m)*kronecker(m,7)*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 + 2*Sum_{n>=1} Fibonacci(n)*Kronecker(n,7)*x^n/(1 - Lucas(n)*(-x)^n + (-1)^n*x^(2*n)).

A209453 a(n) = Pell(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, 54, -45, -1404, 6264, 1890, -76050, 187272, -8865, -1540944, 6200280, -1621620, -51195330, 109055700, 42125400, -868685040, 2946297888, 74093670, -21584605122, 44912353824, -17376284250, -302040439920, 1069478852112, 249392931480, -7095191496489
Offset: 0

Views

Author

Paul D. Hanna, Mar 10 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 + 54*x^2 - 45*x^3 - 1404*x^4 + 6264*x^5 + 1890*x^6 +...
where A(x) = 1 - 1*9*x + 2*27*x^2 - 5*9*x^3 - 12*117*x^4 + 29*216*x^5 + 70*27*x^6 - 169*450*x^7 + 408*459*x^8 +...+ Pell(n)*A109041(n)*^n +...
The g.f. is also given by the identity:
A(x) = 1 - 9*( 1*1*x/(1-2*x-x^2) - 2*4*x^2/(1-6*x^2+x^4) + 12*16*x^4/(1-34*x^4+x^8) - 29*25*x^5/(1-82*x^5-x^10) + 169*49*x^7/(1-478*x^7-x^14) - 408*64*x^8/(1-1154*x^8+x^16) +...).
The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].

Links

Crossrefs

Cf. A109041, A205973, A209452, A209454, A204270, A000129 (Pell), A002203.

Programs

  • Mathematica
    A109041[n_]:= If[n < 1, Boole[n == 0], -9 DivisorSum[n, #^2 KroneckerSymbol[-3, #] &]]; Join[{1}, Table[Fibonacci[n, 2]*A109041[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 - 9*sum(m=1,n,Pell(m)*kronecker(m,3)*m^2*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 - 9*Sum_{n>=1} Pell(n)*Kronecker(n,3)*n^2*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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