Fibonorial

Author

Eric W. Weisstein
December 1, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/Fibonorial.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Fibonorial.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Code

Fibonorial[n_]:=Times@@Fibonacci[Range[n]]
SetAttributes[Fibonorial,Listable]

FibonorialTable[k_]:=Module[{n},
FoldList[Times,1,RecurrenceTable[{a[1]==a[2]==1,a[n]==a[n-1]+a[n-2]},a,{n,k}]]//Rest
]

Eumeration

Fibonorial[Range[10]]

A003266

Asymptotic Behavior

ListPlot[N[Table[(((5^(n/2))Fibonorial[n])/(GoldenRatio^(n(n+1)/2))),{n,100}],100]-QPochhammer[-1/GoldenRatio^2]]

Primes

(Subscript[n!, F])-1

A059709

Position[Fibonorial[Range[100]]-1,_?PrimeQ]//Flatten//Timing

Position[FibonorialTable[100]-1,_?PrimeQ]//Flatten//Timing

Position[Fibonorial[Range[200]]-1,_?PrimeQ]//Flatten//Timing

Position[FibonorialTable[200]-1,_?PrimeQ]//Flatten//Timing

Position[Fibonorial[Range[300]]-1,_?PrimeQ]//Flatten//Timing

Grid[MapIndexed[{#2[[1]],#,IntegerLength[Fibonorial[#]]}&,{4,5,6,7,8,14,15}],Dividers->All,Alignment->Decimal]

(Subscript[n!, F])+1

A053408

Position[Fibonorial[Range[100]]+1,_?PrimeQ]//Flatten//Timing

Position[Fibonorial[Range[200]]+1,_?PrimeQ]//Flatten//Timing

Position[Fibonorial[Range[300]]+1,_?PrimeQ]//Flatten//Timing

Grid[MapIndexed[{#2[[1]],#,IntegerLength[Fibonorial[#]]}&,{1,2,3,4,5,6,7,8,22,28}],Dividers->All,Alignment->Decimal]