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Nexus Number

A Figurate Number built up of the nexus of cells less than $n$ steps away from a given cell. In $k$-D, the $(n+1)$th nexus number is given by

N_{n+1}(k)=\sum_{i=0}^{k} {k\choose i}n^i,

where ${n\choose n}$ is a Binomial Coefficient. The first few $k$-dimensional nexus numbers are given in the table below.
$k$ $N_{n+1}$ name
0 1 unit
1 $1+2n$ Odd Number
2 $1+3n+3n^2$ Hex Number
3 $1+4n+6n^2+4n^3$ Rhombic Dodecahedral Number

See also Hex Number, Odd Number, Rhombic Dodecahedral Number


Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 53-54, 1996.

© 1996-9 Eric W. Weisstein