Centered Triangular Number

A Centered Polygonal Number consisting of a central dot with three dots around it, and then additional dots in the gaps between adjacent dots. The general term is $(3n^2-3n+2)/2$, and the first few such numbers are 1, 4, 10, 19, 31, 46, 64, ... (Sloane's A005448). The Generating Function giving the centered triangular numbers is

$${x(x^2+x+1)\over(1-x)^3}=x+4x^2+10x^3+19x^4+\ldots.$$

See also Centered Pentagonal Number, Centered Square Number

References

Sloane, N. J. A. Sequence A005448/M3378 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.