Problem Archive

A sequence $(a_n)_{n \ge 0}$ starts with $a_0 = 3$ and for each $n \ge 0$,

• if $a_n$ is a triangle numberA triangle number is a number of the form $m(m + 1)/2$ for some integer $m$., then $a_{n + 1} = a_n + 1$;
• otherwise, $a_{n + 1} = 2a_n - a_{n - 1} + 1$.

The sequence begins: $${\color{red}3}, 4, {\color{red}6}, 7, 9, 12, 16, {\color{red}21}, 22, 24, 27, 31, {\color{red}36}, 37, 39, 42, \dots$$ where triangle numbers are marked red.

The $10$th triangle number in the sequence is $a_{2964} = 1439056$.
Find the index $n$ such that $a_n$ is the $70$th triangle number in the sequence.