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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 (list; graph; refs; listen; history; text; internal format)
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 edge-weight 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 edge-weight is assigned as F(2). For k>1, T(k) is obtained from T(k-1) by attaching F(k) vertices to pendents in T(k-1) except r. In T(k), r is retained as in T(k-1) and all the new edge-weights 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, for k>1 D(k)-D(k-1) is this sequence.

Retaining the notation of D(k) above it follows, for k>1 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

Charles R Greathouse IV, Table of n, a(n) for n = 1..97

Eric Weisstein's World of Mathematics, Fibonacci Factorial Constant

FORMULA

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

a(n) = prod(k=1..n, F(k)) * (F(n+2)-1). - Franklin T. Adams-Watters, Jun 23 2011.

PROG

(PARI) s=0; p=1; for(n=1, 40, f=fibonacci(n); s+=f; p*=f; print1(s*p", ")) \\ Charles R Greathouse IV, Jun 21 2011

(PARI) a(n)=prod(k=1, n, fibonacci(k))*(fibonacci(n+2)-1) /* Franklin T. Adams-Watters, Jun 23 2011. */

CROSSREFS

Sequence in context: A320343 A002856 A093461 * A153524 A153552 A295199

Adjacent sequences:  A191991 A191992 A191993 * A191995 A191996 A191997

KEYWORD

nonn

AUTHOR

K.V.Iyer, Venkata Subba Reddy P., Charles R Greathouse IV, Jun 21 2011

STATUS

approved

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Last modified May 15 04:46 EDT 2021. Contains 343909 sequences. (Running on oeis4.)