Squared Triangular Number - Numeric Values; Geometric and Probabilistic Interpretation

Numeric Values; Geometric and Probabilistic Interpretation

The sequence of squared triangular numbers is

0, 1, 9, 36, 100, 225, 441, 784, 1296, 2025, 3025, 4356, 6084, 8281, ... (sequence A000537 in OEIS).

These numbers can be viewed as figurate numbers, a four-dimensional hyperpyramidal generalization of the triangular numbers and square pyramidal numbers.

As Stein (1971) observes, these numbers also count the number of rectangles with horizontal and vertical sides formed in an n×n grid. For instance, the points of a 4×4 grid (or a square made up of 3 smaller squares on a side) can form 36 different rectangles. The number of squares in a square grid is similarly counted by the square pyramidal numbers.

The identity also admits a natural probabilistic interpretation as follows. Let be four integer numbers independently and uniformly chosen at random between 1 and Then, the probability that be not less than any other is equal to the probability that both be not less than and be not less than that is, Indeed, these probabilities are respectively the left and right sides of the Nichomacus identity, normalized over