Odd Number

An Integer of the form $N=2n+1$, where $n$ is an Integer. The odd numbers are therefore ..., $-3$, $-1$, 1, 3, 5, 7, ... (Sloane's A005408), which are also the Gnomic Numbers. The Generating Function for the odd numbers is

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

Since the odd numbers leave a remainder of 1 when divided by two, $N\equiv 1\ \left({{\rm mod\ } {2}}\right)$ for odd $N$. Integers which are not odd are called Even.

See also Even Number, Gnomic Number, Nicomachus's Theorem, Odd Number Theorem, Odd Prime

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.