## Triangular numbers between consecutive square numbers

Let   $S_{m} \: (n)$   be the n-th m-gonal number,   m-gonal number with index   $n$

$S_3 (n) \; = \; n \,(n + 1)/2$   ……….   $S_4 (n) \; = \; n^2$

$S_3 \: (n+1) \; = \; S_3 \: (n) \; + \; (n + 1)$
$S_4 \: (n+1) \; = \; S_4 \: (n) \; + \; (2 \,n + 1)$

Checking up to 50 the following proposition:

If there are exactly two triangular numbers between   $S_4 \: (a+1)$   and   $S_4 \: (a+2)$,
where   $a > 0$,
there is exactly one triangular number between   $S_4 \: (a)$   and   $S_4 \: (a+1)$,   and exactly one
triangular number between   $S_3 \: (a+2)$   and   $S_4 \: (a+3)$

$1^2 = 1$
…………    $T_2 = 3$
$2^2 = 4$
…………    $T_3 = 6$
$3^2 = 9$
…………    $T_4 = 10$
…………    $T_5 = 15$
$4^2 = 16$
…………    $T_6 = 21$
$5^2 = 25$
…………    $T_7 = 28$

$6^2 = 36 = T_8$

…………    $T_9 = 45$
$7^2 = 49$
…………    $T_{10} = 55$
$8^2 = 64$
…………    $T_{11} = 66$
…………    $T_{12} = 78$
$9^2 = 81$
…………    $T_{13} = 91$
$10^2 = 100$
…………    $T_{14} = 105$
…………    $T_{15} = 120$
$11^2 = 121$
…………    $T_{16} = 136$
$12^2 = 144$
…………    $T_{17} = 153$
$13^2 = 169$
…………    $T_{18} = 171$
…………    $T_{19} = 190$
$14^2 = 196$
…………    $T_{20} = 210$

$15^2 = 225$
…………    $T_{21} = 231$
…………    $T_{22} = 253$
$16^2 = 256$
…………    $T_{23} = 276$
$17^2 = 289$
…………    $T_{24} = 300$
$18^2 = 324$
…………    $T_{25} = 325$
…………    $T_{26} = 351$
$19^2 = 361$
…………    $T_{27} = 378$
$20^2 = 400$
…………    $T_{28} = 406$
…………    $T_{29} = 435$
$21^2 = 441$
…………    $T_{30} = 465$
$22^2 = 484$
…………    $T_{31} = 496$
…………    $T_{32} = 528$
$23^2 = 529$
…………    $T_{33} = 561$
$24^2 = 576$
…………    $T_{34} = 595$
$25^2 = 625$
…………    $T_{35} = 630$
…………    $T_{36} = 666$
$26^2 = 676$
…………    $T_{37} = 703$
$27^2 = 729$
…………    $T_{38} = 741$
…………    $T_{39} = 780$
$28^2 = 784$
…………    $T_{40} = 820$
$29^2 = 841$
…………    $T_{41} = 861$
$30^2 = 900$
…………    $T_{42} = 903$
…………    $T_{43} = 946$
$31^2 = 961$
…………    $T_{44} = 990$
$32^2 = 1024$
…………    $T_{45} = 1035$
…………    $T_{46} = 1081$
$33^2 = 1089$
…………    $T_{47} = 1128$
$34^2 = 1156$
…………    $T_{48} = 1176$

$35^2 = 1225 = T_{49}$

…………    $T_{50} = 1275$
$36^2 = 1296$
…………    $T_{51} = 1326$
$37^2 = 1369$
…………    $T_{52} = 1378$
…………    $T_{53} = 1431$
$38^2 = 1444$
…………    $T_{54} = 1485$
$39^2 = 1521$
…………    $T_{55} = 1540$
…………    $T_{56} = 1596$
$40^2 = 1600$
…………    $T_{57} = 1653$
$41^2 = 1681$
…………    $T_{58} = 1711$
$42^2 = 1764$
…………    $T_{59} = 1770$
$43^2 = 1849$
…………    $T_{60} = 1830$
…………    $T_{61} = 1891$
$44^2 = 1936$
…………    $T_{62} = 1953$
…………    $T_{63} = 2016$
$45^2 = 2025$
…………    $T_{64} = 2080$
$46^2 = 2116$
…………    $T_{65} = 2145$
$47^2 = 2209$
…………    $T_{66} = 2211$
…………    $T_{67} = 2278$
$48^2 = 2304$
…………    $T_{68} = 2346$
$49^2 = 2401$
…………    $T_{69} = 2415$
…………    $T_{70} = 2485$
$50^2 = 2500$
…………    $T_{71} = 2556$
$51^2 = 2601$

Does this pattern continue?
Explain.