%I #8 Mar 12 2022 13:11:09

%S 1,7,16,47,121,322,841,2207,5776,15127,39601,103682,271441,710647,

%T 1860496,4870847,12752041,33385282,87403801,228826127,599074576,

%U 1568397607,4106118241,10749957122,28143753121,73681302247,192900153616,505019158607

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

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

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

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

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

%e exp(L(x)) = 1 + x + 2^2*x^2 + 3^2*x^3 + 5^2*x^4 + 8^2*x^5 +...

%o (PARI) {a(n)=(fibonacci(n-1)+fibonacci(n+1))^2-2*((n-1)%2)}

%o (PARI) {a(n)=polcoeff(deriv(log(sum(m=0,n,fibonacci(m)^2*x^m)+x*O(x^n))),n)}

%o (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)}

%Y Cf. A007598, A000032, A000045.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Nov 24 2010