Centered polygonal number

A centered Polygonalzahl is a number to which a regular polygon (polygon ) can be put in a specific pattern and a corresponding number of stones. The laying pattern begins with a single stone in the center of the polygon. Around this center stone more polygons are added, wherein the side lengths increase from the inside outwards in each case by one. Depending on the number of pages we speak for example of centered triangular numbers, square numbers centered, centered pentagonal, centered Sechseckszahlen, etc The following pictures show some examples.

The 19 is the fourth -centered triangular number.

The 25 is the fourth -centered square number.

The 31 is the fourth -centered Fünfeckszahl.

The 37 is the fourth -centered Sechseckszahl.

Because of their relationship with a geometric figure are the centered Polygonalzahlen to the class of figurate numbers. Another way to recycle numbers on polygons represent the ( decentralized ) Polygonalzahlen dar.

Calculation

The - te - centered Eckszahl calculated according to the formula

Alternatively, the - te - centered Eckszahl also use the -th triangular number according to the formula

Calculate.