A121708 Numerator of Sum/Product of first n Fibonacci numbers A000045[n].

1, 2, 2, 7, 2, 1, 11, 3, 11, 1, 29, 47, 29, 1, 19, 41, 19, 1, 199, 23, 199, 1, 521, 281, 521, 1, 31, 2207, 31, 1, 3571, 107, 3571, 1, 9349, 2161, 9349, 1, 211, 13201, 211, 1, 64079, 1103, 64079, 1, 15251, 90481, 15251, 1, 5779, 14503, 5779, 1, 1149851, 2521

Offset: 1

Author

Alexander Adamchuk, Aug 16 2006, Sep 21 2006

Keywords

• frac
• nonn

Comments

a(1) = 1 and a(4k+2) = 1 for k>0.

For k >1 a(4k-1) = a(4k+1) = A072183(2k+1) = A061447(2k+1) Primitive part of Lucas(n).

Crossrefs

Cf. A000045, A000032, A079451, A001578, A096362, A058036, A121709, A090585.

Cf. A061447, A072183.

Programs

Mathematica

Table[Numerator[Sum[Fibonacci[k],{k,1,n}]/Product[Fibonacci[k],{k,1,n}]],{n,1,100}]
With[{fibs=Fibonacci[Range[60]]},Numerator[Accumulate[fibs]/Rest[ FoldList[ Times,1,fibs]]]] (* This is significantly faster than the first program above *) (* Harvey P. Dale, Aug 19 2012 *)

Formula

a(n) = numerator( sum(k=1..n, Fibonacci(k)) / prod(k=1..n, Fibonacci(k)) ).