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Rhombic Dodecahedral Number


A figurate number which is constructed as a centered cube with a square pyramid appended to each face,

RhoDod_n=CCub_n+6P_(n-1)^((4))
(1)
=(2n-1)(2n^2-2n+1),
(2)

where CCub_n is a centered cube number and P_n^((4)) is a square pyramidal number. The first few are 1, 15, 65, 175, 369, 671, ... (OEIS A005917). The generating function of the rhombic dodecahedral numbers is

 (x(x+1)(x^2+10x+1))/((x-1)^4)=x+15x^2+65x^3+175x^4+....
(3)
HauyRhombicDodecahedron5
HauyRhombicDodecahedron9

A related set of numbers is the number of cubes in the Haűy construction of the rhombic dodecahedron, given by

 HauyRhoDod_k=k^3+6sum_(i=1,3,...,k-2)i^2,
(4)

for k an odd number. Re-indexing with k=2n-1 then gives

 HauyRhoDod_n=(2n-1)(8n^2-14n+7),
(5)

giving the first few values 1, 33, 185, 553, 1233, ... (OEIS A046142).


See also

Escher's Solid, Haűy Construction, Octahedral Number, Rhombic Dodecahedron

Explore with Wolfram|Alpha

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 53-54, 1996.Sloane, N. J. A. Sequences A005917/M4968 and A046142 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Rhombic Dodecahedral Number

Cite this as:

Weisstein, Eric W. "Rhombic Dodecahedral Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RhombicDodecahedralNumber.html

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