Centered pentagonal number

The centered pentagonal numbers belong to the centered polygonal numbers , that is, they are two-dimensional figured numbers . They indicate the number of stones with which it is possible to lay out a pentagon as shown in the diagram opposite.

construction

There is a stone in the middle and more stones are then gradually placed around it, one after the other 5, 10, 15 etc., so that a pentagon is created.

The first centered pentagonal numbers are

1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, 391, 456, ... (sequence A005891 in OEIS )

For some authors, the 0 also counts as the zeroth figured number.

calculation

The -th centered pentagonal number is ${\ displaystyle n}$

{\ displaystyle {\ frac {5 \ cdot (n-1) ^ {2} +5 \ cdot (n-1) +2} {2}} = {\ frac {5n ^ {2} -5n + 2} {2}},}

if one defines 1 as the first centered pentagonal number.

additional

Generating function

The sequence of the centered pentagonal numbers have a generating function , namely

{\ displaystyle {\ frac {x ^ {2} + 3x + 1} {(1-x) ^ {3}}} = \ mathbf {1} + \ mathbf {6} x + \ mathbf {16} x ^ { 2} + \ mathbf {31} x ^ {3} + \ mathbf {51} x ^ {4} + \ ldots}

Related figured numbers

The (decentralized) pentagonal numbers relate to another way of laying stones into pentagons:

The fourth decentralized pentagonal number 22.

The fourth centered pentagonal number 31.

If you do not lay pentagons according to this pattern, but triangle, square or hexagons you get the other centered polygonal numbers or polygonal numbers .

Web links

Eric W. Weisstein: Centered Polygonal Numbers on MathWorld .