Pentagonal Number

Author

Eric W. Weisstein
September 29, 2007

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegerSequences/PentagonalNumber.nb.

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

©2007 Wolfram Research, Inc. except for portions noted otherwise

Definition

PentagonalNumber[n]

PentagonalNumber[Range[15]]

Figure

With[{n=5},
Show[GraphicsArray[Graphics[#,AspectRatio->Automatic,PlotRange->All]&/@Table[PolygonalDiagram[n,m,PointStyle->{PointSize[.1],Red}],{m,5}]]]
]

Representation as a Sum of Pentagonal Numbers

NotRepresentable[l_,n_]:=Complement[Range[0,Last[l]],Select[Union[Total/@Subsets[Flatten[Transpose[Table[l,{n}]]],{n}]],#≤Last[l]&]]

3

NotRepresentable[PentagonalNumber[Range[0,150]],3]//Timing

Length[Last[%]]

4

NotRepresentable[PentagonalNumber[Range[0,30]],4]//Timing

Length[Last[%]]

5

NotRepresentable[PentagonalNumber[Range[0,10]],5]//Timing

Generating Function

<<DiscreteMath`RSolve`

gf=FullSimplify[PowerSum[PentagonalNumber[n],{x,n}]]

Series[gf,{x,0,5}]

Congruence Generator

l=Select[Range[100],GCD[#,6]==1&]

(#^2-1)/24&/@l

l2=Select[l,Mod[#,6]==5&]

(#^2-1)/24&/@l2