Square of an odd number

Yes. If $a$ is an odd number, by definition, it can be written as $a=2b+1$ where $b$ is an integer.
So its square is given by:
$$a^{2}=(2b+1)^{2}=(2b)^{2}+2\times 2b+1^{2}=4b^{2}+4b+1$$and this can be written:
$$a^{2}=2\times(2b^{2}+2b)+1.$$In other words, it is of the form $a^{2}=2c+1$, so $a^{2}$ is an odd number.