Noble Number

Author

Eric W. Weisstein
April 20, 2004

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

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

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

Code

NobleNumber[a_List]:=FromContinuedFraction[Append[Prepend[a,0],{1}]]
NobleNumber2[l_List]:=Module[{an,bn,an1,bn1},
{an,bn}=Through[{Numerator,Denominator}[FromContinuedFraction[Prepend[l,0]]]];
{an1,bn1}=Through[{Numerator,Denominator}[FromContinuedFraction[Prepend[Most[l],0]]]];
(an+ an1/GoldenRatio)/(bn+bn1/GoldenRatio)
]

Examples

[0,(Subscript[a, 1]),(OverBar[1])]

FullSimplify[NobleNumber[{(Subscript[a, 1])}]]

Table[NobleNumber[{i}],{i,5}]//FullSimplify

FullSimplify[Table[NobleNumber2[{i}],{i,5}]//FunctionExpand]

[0,(Subscript[a, 1]),(Subscript[a, 2]),(OverBar[1])]

Simplify[NobleNumber[{(Subscript[a, 1]),(Subscript[a, 2])}]]

Table[NobleNumber[{i,j}],{i,5},{j,5}]

FullSimplify[Table[NobleNumber2[{i,j}],{i,5},{j,5}]//FunctionExpand]