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 1-6 of 6 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).

A205967 a(n) = Fibonacci(n)*A008653(n) for n>=1, with a(0)=1, where A008653 is the theta series of direct sum of 2 copies of hexagonal lattice.

Original entry on oeis.org

1, 12, 36, 24, 252, 360, 288, 1248, 3780, 408, 11880, 12816, 12096, 39144, 108576, 43920, 367164, 344952, 93024, 1003440, 3409560, 1050816, 7651152, 8253216, 8346240, 27909300, 61182072, 2357016, 213568992, 185122440, 179720640, 516967296, 1646801604, 507539232
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2012

Keywords

Comments

Compare g.f. to the Lambert series of A008653: 1 + 12*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 + 12*x + 36*x^2 + 24*x^3 + 252*x^4 + 360*x^5 + 288*x^6 +...
where A(x) = 1 + 1*12*x + 1*36*x^2 + 2*12*x^3 + 3*84*x^4 + 5*72*x^5 + 8*36*x^6 +...+ Fibonacci(n)*A008653(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 12*( 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. A008653, A205966, A205968, A203847, A000204 (Lucas).
Cf. A209447 (Pell variant).

Programs

  • Mathematica
    terms = 34; s = 1 + 12*Sum[Fibonacci[n]*KroneckerSymbol[n, 3]^2*n*(x^n/(1 - LucasL[n]*x^n + (-1)^n*x^(2*n))), {n, 1, terms}] + O[x]^terms; CoefficientList[s, x] (* Jean-François Alcover, Jul 05 2017 *)
b[n_] := If[n < 1, Boole[n == 0], 12 Sum[If[Mod[d, 3] > 0, d, 0], {d, Divisors@n}]]; Table[If[n == 0, 1, b[n]*Fibonacci[n]], {n, 0, 50}] (* G. C. Greubel, Jul 17 2018 *)
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 + 12*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=0,50,print1(a(n),", "))

Formula

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

A205969 a(n) = Fibonacci(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, 4, -4, 6, 0, 32, -52, 84, -68, 0, 0, 288, -932, 3016, 0, 1974, 0, 10336, -16724, 0, -43784, 0, 0, 185472, -150050, 971144, -392836, 1271244, 0, 0, -5385076, 8713236, 0, 0, 0, 29860704, -96631268, 312705352, -252983944, 0, 0, 2143314368, -1733977748, 0
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 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 + 4*x^2 - 4*x^3 + 6*x^4 + 32*x^6 - 52*x^7 + 84*x^8 +...
where A(x) = 1 - 1*2*x + 1*4*x^2 - 2*2*x^3 + 3*2*x^4 + 8*4*x^6 - 13*4*x^7 + 21*4*x^8 +...+ Fibonacci(n)*A113973(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) + 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. A113973, A205966, A205968, A205970, A203847, A000204 (Lucas).
Cf. A209449 (Pell variant).

Programs

  • Mathematica
    A113973:= CoefficientList[Series[EllipticTheta[3, q^3]^3/EllipticTheta[3, 0, q], {q, 0, 75}], q]; Table[If[n == 1, 1, Fibonacci[n-1]*A113973[[n]] ], {n, 1, 50}] (* G. C. Greubel, Jul 17 2018 *)
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 - 2*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=0,60,print1(a(n),", "))

Formula

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

A209446 a(n) = Pell(n)*A004016(n) for n >= 1, with a(0)=1, where A004016(n) is the number of integer solutions (x,y) to x^2 + x*y + y^2 = n.

Original entry on oeis.org

1, 6, 0, 30, 72, 0, 0, 2028, 0, 5910, 0, 0, 83160, 401532, 0, 0, 2824992, 0, 0, 79501308, 0, 463367580, 0, 0, 0, 7870428726, 0, 45872220270, 221490672624, 0, 0, 3116610274188, 0, 0, 0, 0, 127800022137480, 617073093431772, 0, 3596565555708780, 0, 0, 0, 122177355889216668
Offset: 0

Views

Author

Paul D. Hanna, Mar 10 2012

Keywords

Comments

Compare g.f. to the Lambert series of A004016: 1 + 6*Sum_{n>=1} Kronecker(n,3)*x^n/(1 - x^n).

Examples

			G.f.: A(x) = 1 + 6*x + 30*x^3 + 72*x^4 + 2028*x^7 + 5910*x^9 + 83160*x^12 + ...
where A(x) = 1 + 1*6*x + 5*6*x^3 + 12*6*x^4 + 169*12*x^7 + 985*6*x^9 + 13860*6*x^12 + ... + Pell(n)*A004016(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) + ...).
The values of the symbol Kronecker(n,3) repeat [1, -1, 0, ...].

Links

Crossrefs

Cf. A004016, A205966, A209445, A209447, A204270, A000129 (Pell), A002203.

Programs

  • Mathematica
    A004016[n_]:= If[n < 1, Boole[n == 0], 6 DivisorSum[n, KroneckerSymbol[#, 3] &]]; Join[{1}, Table[Fibonacci[n, 2]*A004016[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 + 6*sum(m=1,n,kronecker(m,3)*Pell(m)*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 + 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).

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)).

A205965 a(n) = Fibonacci(n)*A001227(n) for n>=1, where A001227(n) is the number of odd divisors of n.

Original entry on oeis.org

1, 1, 4, 3, 10, 16, 26, 21, 102, 110, 178, 288, 466, 754, 2440, 987, 3194, 7752, 8362, 13530, 43784, 35422, 57314, 92736, 225075, 242786, 785672, 635622, 1028458, 3328160, 2692538, 2178309, 14098312, 11405774, 36909860, 44791056, 48315634, 78176338, 252983944
Offset: 1

Views

Author

Paul D. Hanna, Feb 03 2012

Keywords

Comments

Compare g.f. to the Lambert series of A001227: Sum_{n>=1} x^(2*n-1)/(1 - x^(2*n-1)).

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 3*x^4 + 10*x^5 + 16*x^6 + 26*x^7 + 21*x^8 +...
where A(x) = 1*1*x + 1*1*x^2 + 2*2*x^3 + 3*1*x^4 + 5*2*x^5 + 8*2*x^6 + 13*2*x^7 + 21*1*x^8 +...+ Fibonacci(n)*A001227(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1*x/(1-x-x^2) + 2*x^3/(1-4*x^3-x^6) + 5*x^5/(1-11*x^5-x^10) + 13*x^7/(1-29*x^7-x^14) + 34*x^9/(1-76*x^9-x^18) + 89*x^11/(1-199*x^11-x^22) +...
which involves odd-indexed Fibonacci and Lucas numbers.

Links

Crossrefs

Cf. A001227, A205964, A205966, A203847, A000204 (Lucas).
Cf. A209445 (Pell variant).

Programs

  • Mathematica
    A001227[n_]:= DivisorSum[n, Mod[#, 2] &]; Table[A001227[n]*Fibonacci[n], {n, 1, 50}] (* G. C. Greubel, Jul 17 2018 *)
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1,n,fibonacci(2*m-1)*x^(2*m-1)/(1-Lucas(2*m-1)*x^(2*m-1)-x^(4*m-2)+x*O(x^n))),n)}
for(n=1,40,print1(a(n),", "))
  • PARI
    a(n) = fibonacci(n)*sumdiv(n, d, d%2); \\ Michel Marcus, Jul 18 2018

Formula

G.f.: Sum_{n>=1} Fibonacci(2*n-1)*x^(2*n-1)/(1 - Lucas(2*n-1)*x^(2*n-1)-x^(4*n-2)).
Showing 1-6 of 6 results.

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