Square pyramidal number

The Square Pyramidalzahlen among the figurate numbers, more specifically to the Pyramidalzahlen. You quantify the numbers of bullets with which you can build a pyramid square base. As the following illustration shows it at the example of the fourth square Pyramidalzahl 30, they are the sums of the first square numbers.

In the following, denote the -th square Pyramidalzahl.

The following applies:

The first square Pyramidalzahlen are

For some authors, the zero is not a square Pyramidalzahl, so the numbers only begin with the one.

Generating function

The generating function of the square Pyramidalzahlen is:

Relations to other figurate numbers, other representations

It is

With the binomial coefficients and

With the tetrahedral numbers

Also, applies with the -th triangular number:

Related figured numbers

  • The other Pyramidalzahlen, for example, the tetrahedral numbers.
  • The sum of two consecutive square Pyramidalzahlen is a Oktaederzahl.

Others

  • 4900 is in addition to the trivial case 1, the only number that is a square number and a square at the same time Pyramidalzahl. This was proved by G. N. Watson, 1918.
  • The sum of the reciprocals of all quadratic Pyramidalzahlen is:

Derivation of the empirical formula

The difference between two consecutive square numbers is always an odd number. More precisely, due to the difference between the k th and (k-1 )-th square number 2 k -1 is. This yields the following scheme:

A square number can thus be represented as a sum of odd numbers, ie it applies. This sum representation will be used to represent the sum of the first n square numbers by arranged to form a triangle amount of odd numbers. The sum of all odd numbers occurring in the triangle corresponds exactly to the sum of the first n square numbers.

Now you can arrange the same odd numbers still two other ways to a congruent triangle.

These triangles you put now above the other, then the sum of each column consisting of 3 numbers is always constant and there are those columns. Thus the sum of all odd numbers of three triangles and this is exactly three times the sum of the first n square numbers. Thus:

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