Centered Square Number

A Centered Polygonal Number consisting of a central dot with four dots around it, and then additional dots in the gaps between adjacent dots. The general term is $n^2+(n-1)^2$, and the first few such numbers are 1, 5, 13, 25, 41, ... (Sloane's A001844). Centered square numbers are the sum of two consecutive Square Numbers and are congruent to 1 (mod 4). The Generating Function giving the centered square numbers is

$${x(x+1)^2\over(1-x)^3}=x+5x^2+13x^3+25x^4+\ldots.$$

See also Centered Pentagonal Number, Centered Polygonal Number, Centered Triangular Number, Square Number

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 41, 1996.

Sloane, N. J. A. Sequence A001844/M3826 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.