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.

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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)).

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)).

A203838 a(n) = sigma_3(n)*Fibonacci(n), where sigma_3(n) = A001158(n), the sum of cubes of divisors of n.

Original entry on oeis.org

1, 9, 56, 219, 630, 2016, 4472, 12285, 25738, 62370, 118548, 294336, 512134, 1167192, 2152080, 4620147, 7847658, 17604792, 28681660, 62224470, 105431872, 212319468, 348698376, 759507840, 1181718775, 2401396326, 4014783920, 7980869832, 12542045310
Offset: 1

Views

Author

Paul D. Hanna, Jan 12 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 + 9*x^2 + 56*x^3 + 219*x^4 + 630*x^5 + 2016*x^6 +...
where A(x) = x/(1-x-x^2) + 2^3*1*x^2/(1-3*x^2+x^4) + 3^3*2*x^3/(1-4*x^3-x^6) + 4^3*3*x^4/(1-7*x^4+x^8) + 5^3*5*x^5/(1-11*x^5-x^10) + 6^3*8*x^6/(1-18*x^6+x^12) +...+ n^3*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...

Crossrefs

Cf. A203847, A203848, A203849, A001158 (sigma_3), A000204 (Lucas), A000045.

Programs

  • PARI
    {a(n)=sigma(n,3)*fibonacci(n)}
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1,n,m^3*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^3*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_3(n)*fibonacci(n)*x^n, where Lucas(n) = A000204(n).

A204060 G.f.: Sum_{n>=1} Fibonacci(n^2)*x^(n^2).

Original entry on oeis.org

1, 0, 0, 3, 0, 0, 0, 0, 34, 0, 0, 0, 0, 0, 0, 987, 0, 0, 0, 0, 0, 0, 0, 0, 75025, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14930352, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7778742049, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10610209857723, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 37889062373143906
Offset: 1

Views

Author

Paul D. Hanna, Jan 12 2012

Keywords

Comments

Compare g.f. to the Lambert series identity: Sum_{n>=1} lambda(n)*x^n/(1-x^n) = Sum_{n>=1} x^(n^2).
Liouville's function lambda(n) = (-1)^k, where k is number of primes dividing n (counted with multiplicity).

Examples

			G.f.: A(x) = x + 3*x^4 + 34*x^9 + 987*x^16 + 75025*x^25 + 14930352*x^36 +...
where A(x) = x/(1-x-x^2) + (-1)*1*x^2/(1-3*x^2+x^4) + (-1)*2*x^3/(1-4*x^3-x^6) + (+1)*3*x^4/(1-7*x^4+x^8) + (-1)*5*x^5/(1-11*x^5-x^10) + (+1)*8*x^6/(1-18*x^6+x^12) +...+ lambda(n)*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...

Crossrefs

Cf. A203847, A054783, A008836 (lambda), A000204 (Lucas), A000045.
Cf. A209614 (variant).

Programs

  • PARI
    {a(n)=issquare(n)*fibonacci(n)}
  • PARI
    {lambda(n)=local(F=factor(n));(-1)^sum(i=1,matsize(F)[1],F[i,2])}
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1,n,lambda(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} lambda(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)), where lambda(n) = A008836(n) and Lucas(n) = A000204(n).

A205963 a(n) = Fibonacci(n)*A000118(n) for n>=1 with a(0)=1, where A000118(n) is the number of ways of writing n as a sum of 4 squares.

Original entry on oeis.org

1, 8, 24, 64, 72, 240, 768, 832, 504, 3536, 7920, 8544, 13824, 26096, 72384, 117120, 23688, 229968, 806208, 668960, 974160, 2802176, 5100768, 5502144, 4451328, 18606200, 40788048, 62853760, 61019712, 123414960, 479255040, 344644864, 52279416, 1353437952, 2463647184
Offset: 0

Views

Author

Paul D. Hanna, Feb 03 2012

Keywords

Comments

Compare g.f. to the Lambert series of A000118: 1 + 8*Sum_{n>=1} n*x^n/(1 + (-x)^n).

Examples

			G.f.: A(x) = 1 + 8*x + 24*x^2 + 64*x^3 + 72*x^4 + 240*x^5 + 768*x^6 +...
where A(x) = 1 + 1*8*x + 1*24*x^2 + 2*32*x^3 + 3*24*x^4 + 5*48*x^5 + 8*96*x^6 + 13*64*x^7 + 21*24*x^8 +...+ Fibonacci(n)*A000118(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 8*( 1*1*x/(1-x-x^2) + 1*2*x^2/(1+3*x^2+x^4) + 2*3*x^3/(1-4*x^3-x^6) + 3*4*x^4/(1+7*x^4+x^8) + 5*5*x^5/(1-11*x^5-x^10) + 8*6*x^6/(1+18*x^6+x^12) + 13*7*x^7/(1-29*x^7-x^14) +...).

Links

Crossrefs

Cf. A000118, A205507, A205964, A203847, A000204 (Lucas).
Cf. A209443 (Pell variant).

Programs

  • Mathematica
    Join[{1}, Table[Fibonacci[n]*SquaresR[4, n], {n,1,50}]] (* G. C. Greubel, Mar 09 2017 *)
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1+8*sum(m=1,n,fibonacci(m)*m*x^m/(1+Lucas(m)*(-x)^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
for(n=0,31,print1(a(n),", "))

Formula

G.f.: 1 + 8*Sum_{n>=1} Fibonacci(n)*n*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)).

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)).

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)).

A205975 a(n) = Fibonacci(n)*A002652(n) for n>=1, with a(0)=1, where A002652 lists the coefficients in theta series of Kleinian lattice Z[(-1+sqrt(-7))/2].

Original entry on oeis.org

1, 2, 4, 0, 18, 0, 0, 26, 168, 68, 0, 356, 0, 0, 1508, 0, 9870, 0, 10336, 0, 0, 0, 141688, 114628, 0, 150050, 0, 0, 1906866, 2056916, 0, 0, 26139708, 0, 0, 0, 89582112, 96631268, 0, 0, 0, 0, 0, 1733977748, 8416904796, 0, 14690495224, 0, 0, 15557484098
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2012

Keywords

Comments

Compare the g.f. to the Lambert series of A002652:
1 + 2*Sum_{n>=1} Kronecker(n,7)*x^n/(1-x^n).

Examples

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 18*x^4 + 26*x^7 + 168*x^8 + 68*x^9 + 356*x^11 +...
where A(x) = 1 + 1*2*x + 1*4*x^2 + 3*6*x^4 + 14*2*x^7 + 21*8*x^8 + 34*2*x^9 + 89*4*x^11 + 377*4*x^14 + 987*10*x^16 +...+ Fibonacci(n)*A002652(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. A002652, A205974, A205976, A203847, A000204 (Lucas).
Cf. A209455 (Pell variant).

Programs

  • Mathematica
    terms = 50; s = 1 + 2 Sum[Fibonacci[n]*KroneckerSymbol[n, 7]*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 *)
  • 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=0,60,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)).

A204291 G.f.: Sum_{n>=1} moebius(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)), where Lucas(n) = A000204(n).

Original entry on oeis.org

1, 0, 1, 0, 4, -3, 12, 0, 17, -10, 88, -54, 232, -28, 184, 0, 1596, -969, 4180, -1230, 4632, -198, 28656, -17388, 60020, -520, 98209, -23604, 514228, -461932, 1346268, 0, 1722688, -3570, 6672168, -5598882, 24157816, -9348, 31351552, -18606210, 165580140
Offset: 1

Views

Author

Paul D. Hanna, Jan 14 2012

Keywords

Comments

Compare g.f. to the identity: x = Sum_{n>=1} moebius(n)*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).

Examples

			G.f.: A(x) = x + x^3 + 4*x^5 - 3*x^6 + 12*x^7 + 17*x^9 - 10*x^10 + 88*x^11 +...
where A(x) = x/(1-x-x^2) - x^2/(1-3*x^2+x^4) - x^3/(1-4*x^3-x^6) - x^5/(1-11*x^5-x^10) + x^6/(1-18*x^6+x^12) +...+ moebius(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...

Links

Crossrefs

Cf. A203847, A000204 (Lucas), A000045, A008683.

Programs

  • Maple
    with(numtheory): seq(add(mobius(d)*combinat[fibonacci](n)/combinat[fibonacci](d), d in divisors(n)), n=1..60); # Ridouane Oudra, Apr 09 2025
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1,n,moebius(m)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}

Formula

a(k) = 0 iff k = 2^n for n>=1.
a(n) = Fibonacci(n) * Sum_{d|n} mu(d)/Fibonacci(d). - Ridouane Oudra, Apr 09 2025
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