Triangular Numbers which are also Square/Sequence

Sequence of Triangular Numbers which are also Square
The sequence of triangular numbers which are also square begins:
 * $1, 36, 1225, 41 \, 616, 1 \, 413 \, 721, \ldots$

Their indices are:
 * $1, 8, 49, 288, 1681, 9800, \ldots$

Their roots are:
 * $1, 6, 35, 204, 1189, \ldots$