Near Noble Number

Author

Eric W. Weisstein
April 20, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumberTheory/NearNobleNumber.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Code

NearNobleNumber[p_,n_]:=FromContinuedFraction[{0,Append[Table[1,{p-1}],n]}]

NearNobleNumber2[p_,n_]:=(1/2)n((Sqrt[1+4 ((n Fibonacci[p-1]+Fibonacci[p-2])/(n^2Fibonacci[p]))])-1)

Table

nn1=Table[NearNobleNumber[p,n],{p,5},{n,5}]

nn2=Table[NearNobleNumber2[p,n],{p,5},{n,5}]//Simplify

Simplify[nn1-nn2]

Special Cases

n=1

n1=FullSimplify[NearNobleNumber2[p,1]]

n=2

n2=FullSimplify[NearNobleNumber2[p,2]]

FullSimplify[Subtract@@{n2,Sqrt[Fibonacci[p+2]/Fibonacci[p]]-1},p∈Integers]

n=3

n3=FullSimplify[NearNobleNumber2[p,3],p∈Integers&&p>0]

Table[Subtract@@{n3,(1/2) (-3+(Sqrt[9+4Fibonacci[2p]/Fibonacci[p]^2]))},{p,5}]

FullSimplify[Subtract@@{n3,(1/2) (-3+(Sqrt[9+4Fibonacci[2p]/Fibonacci[p]^2]))},p∈Integers]

FullSimplify[(1/2) (-3+(Sqrt[9+4Fibonacci[2p]/Fibonacci[p]^2])),p∈Integers&&p>0]//TraditionalForm

n=4

n4=FullSimplify[NearNobleNumber2[p,4],p∈Integers&&p>0]