Doubly triangular number

In mathematics, the doubly triangular numbers are the numbers that appear within the sequence of triangular numbers, in positions that are also triangular numbers.

The doubly triangular numbers form the sequence[1] The

th doubly triangular number is given by the quartic formula[2]

The sums of row sums of Floyd's triangle give the doubly triangular numbers.

Another way of expressing this fact is that the sum of all of the numbers in the first

rows of Floyd's triangle is the

th doubly triangular number.

[1][2] Doubly triangular numbers arise naturally as numbers of unordered pairs of unordered pairs of objects, including pairs where both objects are the same: When pairs with both objects the same are excluded, a different sequence arises, the tritriangular numbers

[5] Some numerologists and biblical studies scholars consider it significant that 666, the number of the beast, is a doubly triangular number.

There are 21 colorings of the four corners of a square using three colors (up to symmetry), a doubly triangular number, formed by combining two of the six colorings of two opposite corners