Square Triangular Number - Explicit Formulas

Explicit Formulas

Write Nk for the kth square triangular number, and write sk and tk for the sides of the corresponding square and triangle, so that

The sequences Nk, sk and tk are the OEIS sequences  A001110,  A001109, and  A001108 respectively.

In 1778 Leonhard Euler determined the explicit formula

N_k = \left( \frac{(3 + 2\sqrt{2})^k - (3 - 2\sqrt{2})^k}{4\sqrt{2}} \right)^2.

Other equivalent formulas (obtained by expanding this formula) that may be convenient include

\begin{align} N_k &= {1 \over 32} \left( ( 1 + \sqrt{2} )^{2k} - ( 1 - \sqrt{2} )^{2k} \right)^2 = {1 \over 32} \left( ( 1 + \sqrt{2} )^{4k}-2 + ( 1 - \sqrt{2} )^{4k} \right) \\ &= {1 \over 32} \left( ( 17 + 12\sqrt{2} )^k -2 + ( 17 - 12\sqrt{2} )^k \right). \end{align}

The corresponding explicit formulas for sk and tk are

and

Read more about this topic:  Square Triangular Number

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