Gnomic Number

A Figurate Number of the form $g_n=2n-1$ which are the areas of square gnomons, obtained by removing a Square of side $n-1$ from a Square of side $n$,

$$g_n=n^2-(n-1)^2=2n-1.$$

The gnomic numbers are therefore equivalent to the Odd Numbers, and the first few are 1, 3, 5, 7, 9, 11, ... (Sloane's A005408). The Generating Function for the gnomic numbers is

$${x(1+x)\over(x-1)^2}=x+3x^2+5x^3+7x^4+\ldots.$$

See also Odd Number

References

Sloane, N. J. A. Sequence A005408/M2400 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.