Triangular Numbers which are also Square

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Theorem

Let $A_n$ be the $n$th non-negative integer whose square is also a triangular number.

Then:

$A_n = \begin{cases} 0 & : n = 0 \\ 1 & : n = 1 \\ 6 A_{n - 1} - A_{n - 2} & : n > 1 \end{cases}$


Sequence of Square Triangles

The sequence of triangular numbers which are also square begins:

$1, 36, 1225, 41 \, 616, 1 \, 413 \, 721, \ldots$

This sequence is A001110 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Their indices are:

$1, 8, 49, 288, 1681, 9800, \ldots$

This sequence is A001108 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Their roots are:

$1, 6, 35, 204, 1189, \ldots$

This sequence is A001109 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof


Sources