Odd numbers

The odd numbers are the integers not divisible by 2. Thus all prime numbers (other than 2) are odd. The odd numbers satisfy the congruences $\scriptstyle n\,\equiv \,1{\pmod {2}}\,$ and $\scriptstyle n\,\equiv \,\pm 1{\pmod {4}}\,$.

The odd numbers are the square gnomonic numbers, i.e.

$2n+1=(n+1)^{2}-n^{2},\,n\geq 0.\,$

The generating function of odd numbers $\scriptstyle 2n-1,\,n\,\geq \,1,\,\,$ is

$G_{\{2n-1,\,n\geq 1\}}(x)\equiv \sum _{n=1}^{\infty }(2n-1){\frac {x^{n}}{n!}}={\frac {x(x+1)}{(x-1)^{2}}}\,$

The generating function of odd numbers $\scriptstyle 2n+1,\,n\,\geq \,0,\,\,$ is

$G_{\{2n+1,\,n\geq 0\}}(x)\equiv \sum _{n=0}^{\infty }(2n+1){\frac {x^{n}}{n!}}={\frac {(x+1)}{(x-1)^{2}}}\,$

The positive odd numbers are listed in A005408.