## Indices of two consecutive Triangular Num3ers

The difference of the indices of two consecutive triangular numbers, each a square, is equal to the sum of two consecutive integers the sum of whose squares is a square.

Checking up the first few,

$T_1 \; = \; 1$

$T_8 \; = \; 36 \; = \; 6^2$
$8 \; - \; 1 \; = \; 7 \; = \; 3 \; + \; 4$
$3^2 \; + \; 4^2 \; = \; 5^2$

$T_{49} \; = \; 1225 \; = \; 35^2$
$49 \; - \; 8 \; = \; 41 \; = \; 20 \; + \; 21$
$20^2 \; + \; 21^2 \; = \; 29^2$

$T_{288} \; = \; 41616 \; = \; 204^2$
$288 \; - \; 49 \; = \; 239 \; = \; 119 \; + \; 120$
$119^2 \; + \; 120^2 \; = \; 169^2$

$T_{1681} \; = \; 1413721 \; = \; 1189^2$
$1681 \; - \; 288 \; = \; 1393 \; = \; 696 \; + \; 697$
$696^2 \; + \; 697^2 \; = \; 985^2$

$T_{9800} \; = \; 48024900 \; = \; 6930^2$
$9800 \; - \; 1681 \; = \; 8119 \; = \; 4059 \; + \; 4060$
$4059^2 \; + \; 4060^2 \; = \; 5741^2$

$T_{57121} \; = \; 1631432881 \; = \; 40391^2$
$57121 \; - \; 9800 \; = \; 47321 \; = \; 23660 \; + \; 23661$
$23660^2 \; + \; 23661^2 \; = \; 33461^2$

$T_{332928} \; = \; 55420693056 \; = \; 235416^2$
$332928 \; - \; 57121 \; = \; 275807 \; = \; 137903 \; + \; 137904$
$137903^2 \; + \; 137904^2 \; = \; 195025^2$

$T_{1940449} \; = \; 1882672131025 \; = \; 1372105^2$
$1940449 \; - \; 332928 \; = \; 1607521 \; = \; 803760 \; + \; 803761$
$803760^2 \; + \; 803761^2 \; = \; 1136689^2$

$T_{11309768} \; = \; 63955431761796 \; = \; 7997214^2$
$11309768 \; - \; 1940449 = 9369319 \; = \; 4684659 \; + \; 4684660$
$4684659^2 \; + \; 4684660^2 \; = \; 6625109^2$