A204060 -id:A204060 - cp's OEIS frontend

A203847 a(n) = tau(n)*Fibonacci(n), where tau(n) = A000005(n), the number of divisors of n.

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

Paul D. Hanna, Jan 11 2012

nonn

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).

			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)) +...

G. C. Greubel, Table of n, a(n) for n = 1..2500

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.

Table[DivisorSigma[0, n]*Fibonacci[n], {n, 50}] (* G. C. Greubel, Jul 17 2018 *)

PARI
```
{a(n)=sigma(n,0)*fibonacci(n)}
```

{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)}

a(n) = numdiv(n)*fibonacci(n); \\ Michel Marcus, Jul 18 2018

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).

A204274 G.f.: Sum_{n>=1} Pell(n^2)*x^(n^2).

Original entry on oeis.org

1, 0, 0, 12, 0, 0, 0, 0, 985, 0, 0, 0, 0, 0, 0, 470832, 0, 0, 0, 0, 0, 0, 0, 0, 1311738121, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21300003689580, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2015874949414289041, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Paul D. Hanna, Jan 14 2012

nonn

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).

			G.f.: A(x) = x + 12*x^4 + 985*x^9 + 470832*x^16 + 1311738121*x^25 +...
where A(x) = x/(1-2*x-x^2) + (-1)*2*x^2/(1-6*x^2+x^4) + (-1)*5*x^3/(1-14*x^3-x^6) + (+1)*12*x^4/(1-34*x^4+x^8) + (-1)*29*x^5/(1-82*x^5-x^10) + (+1)*70*x^6/(1-198*x^6+x^12) +...+ lambda(n)*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) +...

Robert Israel, Table of n, a(n) for n = 1..2500

Cf. A204327, A204060, A204270, A204271, A204272, A204273, A204275, A008836 (lambda), A002203, A000045.

pell:= gfun:-rectoproc({a(0)=0,a(1)=1,a(n)=2*a(n-1)+a(n-2)},a(n),remember):
seq(`if`(issqr(n),pell(n),0), n=1..100); # Robert Israel, Nov 24 2015

CoefficientList[Sum[Fibonacci[n^2, 2] x^n^2/x, {n, 1, 8}], x] (* Jean-François Alcover, Mar 25 2019 *)

/* Subroutines used in PARI programs below: */
{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)}
{A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}

PARI
```
{a(n)=issquare(n)*Pell(n)}
```

{lambda(n)=local(F=factor(n));(-1)^sum(i=1,matsize(F)[1],F[i,2])}
{a(n)=polcoeff(sum(m=1,n,lambda(m)*Pell(m)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}

G.f.: Sum_{n>=1} lambda(n)*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where lambda(n) = A008836(n), Pell(n) = A000129(n) and A002203 is the companion Pell numbers.

A205507 a(n) = Fibonacci(n) * A004018(n) for n>=1 with a(0)=1, where A004018(n) is the number of ways of writing n as a sum of 2 squares.

Original entry on oeis.org

1, 4, 4, 0, 12, 40, 0, 0, 84, 136, 440, 0, 0, 1864, 0, 0, 3948, 12776, 10336, 0, 54120, 0, 0, 0, 0, 900300, 971144, 0, 0, 4113832, 0, 0, 8713236, 0, 45623096, 0, 59721408, 193262536, 0, 0, 818673240, 1324641128, 0, 0, 0, 9079225360, 0, 0, 0, 31114968196

Offset: 0

Paul D. Hanna, Jan 28 2012

nonn

Compare to the g.f. of A004018 given by the Lambert series identity:

1 + 4*Sum_{n>=0} (-1)^n*x^(2*n+1)/(1 - x^(2*n+1)) = (1 + 2*Sum_{n>=1} x^(n^2))^2.

			G.f.: A(x) = 1 + 4*x + 4*x^2 + 12*x^4 + 40*x^5 + 84*x^8 + 136*x^9 + 440*x^10 +...
Compare the g.f to the square of the Jacobi theta_3 series:
theta_3(x)^2 = 1 + 4*x + 4*x^2 + 4*x^4 + 8*x^5 + 4*x^8 + 4*x^9 + 8*x^10 +...+ A004018(n)*x^n +...
The g.f. equals the sum:
A(x) = 1 + 4*x/(1-x-x^2) - 4*2*x^3/(1-4*x^3-x^6) + 4*5*x^5/(1-11*x^5-x^10) - 4*13*x^7/(1-29*x^7-x^14) + 4*34*x^9/(1-76*x^9-x^18) - 4*89*x^11/(1-199*x^11-x^22) + 4*233*x^13/(1-521*x^13-x^26) - 4*610*x^15/(1-1364*x^15-x^30) +...
which involves odd-indexed Fibonacci and Lucas numbers.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A205508, A204060, A203847, A004018, A000204 (Lucas).

Join[{1}, Table[Fibonacci[n]*SquaresR[2, n], {n,1,50}]] (* G. C. Greubel, Mar 05 2017 *)

{A004018(n)=polcoeff((1+2*sum(k=1,sqrtint(n+1),x^(k^2),x*O(x^n)))^2,n)}
{a(n)=if(n==0,1,fibonacci(n)*A004018(n))}

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff((1+4*sum(m=0,n+1,(-1)^m*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)}

G.f.: 1 + 4*Sum_{n>=0} (-1)^n*Fibonacci(2*n+1)*x^(2*n+1) / (1 - Lucas(2*n+1)*x^(2*n+1) - x^(4*n+2)), where Lucas(n) = A000204(n).

A209614 G.f.: Sum_{n>=1} Fibonacci(n^3)*x^(n^3).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 196418, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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

Offset: 1

Paul D. Hanna, Apr 07 2012

nonn

Compare g.f. to the Lambert series identity: Sum_{n>=1} A210826(n)*x^n/(1-x^n) = Sum_{n>=1} x^(n^3), where A210826(n) = kronecker(-3,d(n)) and d(n) is the number of divisors of n.

			G.f.: A(x) = x + 21*x^8 + 196418*x^27 + 10610209857723*x^64 + 59425114757512643212875125*x^125 +...
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) + (0)*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)+ (-1)*13*x^7/(1-29*x^7-x^14) + (+1)*21*x^8/(1-47*x^8+x^16) + (0)*34*x^9/(1-76*x^9-x^18) + (+1)*55*x^10/(1-123*x^10+x^20)+ (-1)*89*x^11/(1-199*x^11-x^22) +...+ A210826(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...
The sequence A210826(n) = kronecker(-3, d(n)) begins:
[1,-1,-1,0,-1,1,-1,1,0,1,-1,0,-1,1,1,-1,-1,0,-1,0,1,1,-1,-1,0,1,1,...].

Cf. A210826, A204060.

{A210826(n)=if(n==0,0,kronecker(-3,numdiv(n)))}
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1, n, A210826(m)*fibonacci(m)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))), n)}
for(n=1,125,print1(a(n),", "))

G.f.: Sum_{n>=1} A210826(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} Fibonacci(n^3)*x^(n^3).

cp's OEIS Frontend

A203847 a(n) = tau(n)*Fibonacci(n), where tau(n) = A000005(n), the number of divisors of n.

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A204274 G.f.: Sum_{n>=1} Pell(n^2)*x^(n^2).

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A205507 a(n) = Fibonacci(n) * A004018(n) for n>=1 with a(0)=1, where A004018(n) is the number of ways of writing n as a sum of 2 squares.

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A209614 G.f.: Sum_{n>=1} Fibonacci(n^3)*x^(n^3).

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