Decagonal number


In mathematics, a decagonal number is a figurate number that extends the concept of triangular and square numbers to the decagon. However, unlike the triangular and square numbers, the patterns involved in the construction of decagonal numbers are not rotationally symmetrical. Specifically, the n-th decagonal numbers counts the dots in a pattern of n nested decagons, all sharing a common corner, where the ith decagon in the pattern has sides made of i dots spaced one unit apart from each other. The n-th decagonal number is given by the following formula
The first few decagonal numbers are:
The nth decagonal number can also be calculated by adding the square of n to thrice the th pronic number or, to put it algebraically, as

Properties

  • Decagonal numbers consistently alternate parity.
  • is the sum of the first natural numbers congruent to 1 mod 8.
  • is number of divisors of.
  • The only decagonal numbers that are square numbers are 0 and 1.
  • The decagonal numbers follow the following recurrence relations:

Sum of reciprocals

The sum of the reciprocals of the decagonal numbers admits a simple closed form:

Proof

This derivation rests upon the method of adding a "constructive zero":
Rearranging and considering the individual sums: