A173661 Logarithmic derivative of the squares of the Fibonacci numbers (A007598, with offset).

1, 7, 16, 47, 121, 322, 841, 2207, 5776, 15127, 39601, 103682, 271441, 710647, 1860496, 4870847, 12752041, 33385282, 87403801, 228826127, 599074576, 1568397607, 4106118241, 10749957122, 28143753121, 73681302247, 192900153616, 505019158607

Offset

1,2

Comments

The Lucas numbers (A000032) forms the logarithmic derivative of the Fibonacci numbers (A000045).

Links

Table of n, a(n) for n=1..28.

Formula

a(n) = Lucas(n)^2 for odd n, a(n) = Lucas(n)^2 - 2 for even n>0.

O.g.f.: x*(1+4*x-5*x^2+2*x^3)/((1-x^2)*(1-3*x+x^2)).

Example

G.f.: L(x) = x + 7*x^2/2 + 16*x^3/3 + 47*x^4/4 + 121*x^5/5 +...
exp(L(x)) = 1 + x + 2^2*x^2 + 3^2*x^3 + 5^2*x^4 + 8^2*x^5 +...

Prog

(PARI) {a(n)=(fibonacci(n-1)+fibonacci(n+1))^2-2*((n-1)%2)}
(PARI) {a(n)=polcoeff(deriv(log(sum(m=0, n, fibonacci(m)^2*x^m)+x*O(x^n))), n)}
(PARI) {a(n)=polcoeff(x*(1+4*x-5*x^2+2*x^3)/((1-x^2)*(1-3*x+x^2+x*O(x^n))), n)}

Crossrefs

Cf. A007598, A000032, A000045.

Sequence in context: A304013 A037241 A154141 * A152530 A065099 A001345

Adjacent sequences: A173658 A173659 A173660 * A173662 A173663 A173664

Keyword

nonn

Author

Paul D. Hanna, Nov 24 2010

Status

approved