

A191994


(Sum of first n Fibonacci numbers) times (product of first n Fibonacci numbers).


2



1, 2, 8, 42, 360, 4800, 102960, 3538080, 196035840, 17520703200, 2529842515200, 590412901478400, 222813349683724800, 136001024583142118400, 134285149587387262464000, 214504624277084224347264000, 554361997358383529330695680000
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OFFSET

1,2


COMMENTS

Let F(1), F(2), F(3), ... be the Fibonacci numbers 1, 1, 2, .... For k=1, we define the tree T(1) the path on two vertices with one identified as the root r. We assign the edgeweight F(1). T(2) is obtained from T(1) by attaching F(2) vertex to the pendents in T(1) except r. In T(2), r is retained as in T(1) and the new edgeweight is assigned as F(2). For k>1, T(k) is obtained from T(k1) by attaching F(k) vertices to pendents in T(k1) except r. In T(k), r is retained as in T(k1) and all the new edgeweights are assigned F(k). With D(1)=1, for k>1 let D(k)=Sum of all distances d(r,x) taken across all vertices x in T(k). By induction it follows that for k>1, D(k)D(k1) is this sequence.
Retaining the notation of D(k) above, it follows, for k>1, that if D(k)=a(1)F(1)+    +a(k)F(k) then D(k+1)=b(1)F(1)+    +b(k)F(k)+b(k+1)F(k+1) where b(k+1) is the number of leaf nodes in T(k+1).


LINKS



FORMULA

a(n) ~ C*sqrt(phi^(n^2 + 3*n + 4)/5^(n+1)) where C = A062073 and phi = (1+sqrt(5))/2.


PROG



CROSSREFS

Cf. A000071 (sum of Fibonacci numbers), A003266 (product of Fibonacci numbers).
Cf. A062073 (Fibonacci factorial constant).


KEYWORD

easy,nonn


AUTHOR



STATUS

approved



