The considered mathematical regularities of the periodic law allow presenting the nucleus charge of an atom (Z) as a function of the period number (p) and number of the element inside this period (n). To illustrate these regularities it is convenient to take the table from laboratory of the Nobel winner Glenn T. Seaborg.
Figure 22. Modified form of a periodic table showing known and predicted electron shells.From G.T. Seaborg, Lawrence Berkeley National Laboratory, 1989
In the Seaborg's table, borrowed from (Britannica), I numbered the periods (p) (white Roman numerals) and posed numbers (n) of elements inside each period (colour Arabic numerals).
The formulas determining dependence of the charge of a nucleus of atom by number of the period and the serial number of an element inside the period
For the odd periods:
Z1 = ( p3 + 3p2 + 5p - 9 ) / 6 + n ;
For the even periods:
Z2 = ( p3 + 3p2 + 2p - 12 ) / 6 + n ,
Z1 - charge of a nucleus of elements of the odd periods;
Z2 - charge of a nucleus of elements of the even periods;
p - number of the period;
n - serial number of an element inside the period.