A203797 a(n) = Pell(n) * Sum_{d|n} 1/Pell(d), where Pell(n) = A000129(n).

1, 3, 6, 19, 30, 120, 170, 647, 1183, 3650, 5742, 24916, 33462, 121652, 240756, 746639, 1136690, 4707147, 6625110, 25882770, 46565244, 139849776, 225058682, 978088748, 1356970471, 4750318586, 9182205852, 29333908544, 44560482150, 188175715440, 259717522850, 994309609247

Offset

1,2

Links

Table of n, a(n) for n=1..32.

Formula

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

Example

G.f.: A(x) = x + 3*x^2 + 6*x^3 + 19*x^4 + 30*x^5 + 120*x^6 + 170*x^7 + ... where A(x) = x/(1-2*x-x^2) + x^2/(1-6*x^2+x^4) + x^3/(1-14*x^3-x^6) + x^4/(1-34*x^4+x^8) + x^5/(1-82*x^5-x^10) + x^6/(1-198*x^6+x^12) + ... + x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) + ...

Mathematica

a[n_] := Fibonacci[n, 2] * DivisorSum[n, 1/Fibonacci[#, 2] &]; Array[a, 32] (* Amiram Eldar, Aug 18 2023 *)

Prog

(PARI) {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)}
{a(n)=Pell(n) * sumdiv(n, d, 1/Pell(d))}
(PARI) /* G.f. using companion Pell numbers: */
{A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}
{a(n)=polcoeff(sum(m=1, n, x^m/(1 - A002203(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))), n)}

Crossrefs

Cf. A204059, A111075, A203802, A000129, A002203.

Sequence in context: A090956 A108972 A304965 * A019097 A347685 A219286

Adjacent sequences: A203794 A203795 A203796 * A203798 A203799 A203800

Keyword

nonn

Author

Paul D. Hanna, Jan 13 2012

Status

approved