Centered Pentagonal Number

A Centered Polygonal Number consisting of a central dot with five dots around it, and then additional dots in the gaps between adjacent dots. The general term is $(5n^2-5n+2)/2$, and the first few such numbers are 1, 6, 16, 31, 51, 76, ... (Sloane's A005891). The Generating Function of the centered pentagonal numbers is

$${x(x^2+3x+1)\over(x-1)^3}=x+6x^2+16x^3+31x^4+\ldots.$$

See also Centered Square Number, Centered Triangular Number

References

Sloane, N. J. A. Sequence A005891/M4112 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.