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

Author

Paul D. Hanna, Nov 24 2010

Keywords

• nonn

Comments

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

Examples

			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 +...
		

Crossrefs

Cf. A007598, A000032, A000045.

Programs

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)}

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)).