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A173661 Logarithmic derivative of the squares of the Fibonacci numbers (A007598, with offset). 0
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 6 23:56 EDT 2020. Contains 335484 sequences. (Running on oeis4.)