Square pyramidal number Totally Explained

Everything about Square Pyramidal Number totally explained

In mathematics, a pyramid number, or square pyramidal number, is a figurate number that represents a pyramid with a base and four sides. These numbers can be expressed in a formula as ť

sum_

where

T_n

is the nth tetrahedral number.
The sum of two consecutive square pyramidal numbers is an octahedral number.
Besides 1, there's only one other number that's both a square and a pyramid number, 4900, the 70th square number and the 24th square pyramidal number. This fact was proven by G. N. Watson in 1918.

Squares in a square

A common mathematical puzzle involves finding the number of squares in a large n by n square grid. This number can be derived as follows:

The number of 1×1 boxes found in the grid is $n^2$ .

The number of 2×2 boxes found in the grid is

(n-1)^2

. These can be counted by counting all of the possible upper-left corners of 2×2 boxes.

The number of k×k boxes (1 ≤ k ≤ n) found in the grid is

(n-k+1)^2

. These can be counted by counting all of the possible upper-left corners of k×k boxes. It follows that the number of squares in an n by n square grid is: ť

x = n^2 + (n-1)^2 + (n-2)^2 + (n-3)^2 + ldots + 1^2

or: ť

x = n(n+1)(2n+1)/6

That is, the solution to the puzzle is given by the square pyramidal numbers.

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