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, 1322157322201, 3461452808002

Offset

1,2

Comments

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

Links

Paolo Xausa, Table of n, a(n) for n = 1..2000

Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

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)).

From Klaus Purath, Sep 15 2025: (Start)

a(n) = A047946(n) - 1.

a(n) = 3*a(n-1) - a(n-2) + 1 + 5*(-1)^n. (End)

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 +...

Mathematica

A173661[n_] := LucasL[n]^2 - (-1)^n - 1; Array[A173661, 30] (* or *)
LinearRecurrence[{3, 0, -3, 1}, {1, 7, 16, 47}, 30] (* Paolo Xausa, Sep 20 2025 *)

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, A047946.

Sequence in context: A304013 A037241 A154141 * A152530 A065099 A001345

Adjacent sequences: A173658 A173659 A173660 * A173662 A173663 A173664

Keyword

nonn,easy

Author

Paul D. Hanna, Nov 24 2010

Status

approved