Rhombic Dodecahedral Number

Author

Eric W. Weisstein
December 22, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Definition

{CenteredCubeNumber[n],PyramidalNumber[n-1,4]}

Factor[CenteredCubicNumber[n]+6PyramidalNumber[n-1,4]]

RhombicDodecahedralNumber[n]

Table

Table[RhombicDodecahedralNumber[n],{n,10}]

Generating Function

Factor[GeneratingFunction[RhombicDodecahedralNumber[n],n,x]]

Series[((-1+5 x+5 (x^2)+15 (x^3))/(((-1+x))^4)),{x,0,3}]

Factor[((-1+5 x+5 (x^2)+15 (x^3))/(((-1+x))^4))+1]

Series[((x (1+x) (1+10 x+(x^2)))/(((-1+x))^4)),{x,0,4}]

Factor/@SeriesCoefficient[((x (1+x) (1+10 x+(x^2)))/(((-1+x))^4)),{x,0,n}]

Hauy

Factor[(k^3)+6Sum[(i^2),{i,1,k-2,2}]]

Factor[%/.k->2n-1]

Table[(-1+2 n) (7-14 n+8 (n^2)),{n,10}]