Pentagonal Number

$p_n = \tfrac{3n^2-n}{2}= \tfrac{3(4)^2-4}{2}=22$

There is a formula in finding the "nth" pentagonal number: Apply the [n(3n - 1)]/2 plan. Since n=4, [4(3*4-1)]/2 [4(11)]/2 44/2 22. Thus, the fourth pentagonal number is 22.

A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside.

pn is given by the formula:

p_n = \tfrac{3n^2-n}{2}

for n ≥ 1. The first few pentagonal numbers are:

1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001 (sequence A000326 in OEIS). [ FROM WIKIPEDIA ]