# Triangular, Square and Pronic Numbers

The triangle numbers are equal to the sum of the $$n$$ natural numbers from $$1$$ to $$n$$.

The sequence begins :

$$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, …$$

$$\displaystyle T_n = \sum_ {k=1}^n k = 1 + 2 + 3 + … +n =\frac {n( n + 1)}{2}$$

So the sum to the tenth natural number $$(10)$$ is :

$$\displaystyle \frac {10 \cdot 11}{2} = 55$$

The square numbers are equal to the sum of the odd $$(2k-1, k = 1, 2, 3, …)$$ numbers from $$1$$ to $$n$$.

The sequence begins :

$$1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …$$

$$\displaystyle S_n = \sum_{k=1}^n (2k-1) = 1 + 3 + 5 + … +2n-1 = n^2$$

So the sum to the tenth odd number $$(19)$$ is :

$$10^2 = 100$$.

The pronic (rectangular) numbers are equal to the sum of the even $$(2k, k = 1, 2, 3, …)$$ numbers from $$1$$ to $$n$$.

The sequence begins :

$$2, 6, 12, 20, 30, 42, 56, 72, 90, 110, …$$

$$\displaystyle P_n = \sum_{k=1}^n 2k = 2 + 4 + 6 + … +2n = 2\sum_{k=1}^n k = n(n + 1)$$

So the sum to the tenth even number $$(20)$$ is :

$$10 \cdot 11 = 110$$.

The sum of two consecutive triangle numbers is a square number.

For example,

$$10 + 15 = 25 = 5^2 = T_4 +T_5 = S_5$$.

$$T_{n-1} + T_n = S_n$$

A pronic number is twice a triangle number.

For example,

$$20 = 2 \cdot 10 = P_4 = 2T_4$$.

$$P_n = 2T_n$$

The sum of a square number and a pronic number is a triangle number.

For example,

$$25 + 30 = 55 = S_5 + P_5 = T_{10}$$.

When the subscript of $$T$$ is even, then

$$S_n +P_n = T_{2n}$$

When the subscript of $$T$$ is odd, then

$$S_n + P_{n-1} = T_{2n-1}$$

© OldTrout $$2018$$

Audio file