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 \(2018\)

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