## Triangular oblongs : a(a+1) = b(b+1)/2

An Oblong number is a number which is the product of two consecutive integers, that is,
a number of the form   $n \,(n + 1)$

A Triangular number is a number of the form   $m \,(m + 1)/2$

A Triangular oblong number:   $n \,(n + 1) \; = \; m \,(m + 1)/2$

Solving the diophantine equation   $2 \, A \, (A + 1) \; = \; B \, (B + 1)$,   we obtain the first few solutions:

$( \, A_0, \; B_0 \, ) \; = \; (0, \; 0)$
$( \, A_1, \; B_1 \, ) \; = \; (2, \; 3)$
$( \, A_2, \; B_2 \, ) \; = \; (14, \; 20)$
$( \, A_3, \; B_3 \, ) \; = \; (84, \; 119)$
$( \, A_4, \; B_4 \, ) \; = \; (492, \; 696)$
$( \, A_5, \; B_5 \, ) \; = \; (2870, \; 4059)$
$( \, A_6, \; B_6 \, ) \; = \; (16730, \; 23660)$
$( \, A_7, \; B_7 \, ) \; = \; (97512, \; 137903)$
$( \, A_8, \; B_8 \, ) \; = \; (568344, \; 803760)$
$( \, A_9, \; B_9 \, ) \; = \; (3312554, \; 4684659)$

$( \, A_{10}, \; B_{10} \, ) \; = \; (19306982, \; 27304196)$
$( \, A_{11}, \; B_{11} \, ) \; = \; (112529340, \; 159140519)$
$( \, A_{12}, \; B_{12} \, ) \; = \; (655869060, \; 927538920)$
$( \, A_{13}, \; B_{13} \, ) \; = \; (3822685022, \; 5406093003)$
$( \, A_{14}, \; B_{14} \, ) \; = \; (22280241074, \; 31509019100)$
$( \, A_{15}, \; B_{15} \, ) \; = \; (129858761424, \; 183648021599)$
$( \, A_{16}, \; B_{16} \, ) \; = \; (756872327472, \; 1070379110496)$
$( \, A_{17}, \; B_{17} \, ) \; = \; (4411375203410, \; 6238626641379)$
$( \, A_{18}, \; B_{18} \, ) \; = \; (25711378892990, \; 36361380737780)$
$( \, A_{19}, \; B_{19} \, ) \; = \; (149856898154532, \; 211929657785303)$
$( \, A_{20}, \; B_{20} \, ) \; = \; (873430010034204, \; 1235216565974040)$
$( \, A_{21}, \; B_{21} \, ) \; = \; (5090723162050694, \; 7199369738058939)$
$( \, A_{22}, \; B_{22} \, ) \; = \; (29670908962269962, \; 41961001862379596)$
$( \, A_{23}, \; B_{23} \, ) \; = \; (172934730611569080, \; 244566641436218639)$
$( \, A_{24}, \; B_{24} \, ) \; = \; (1007937474707144520, \; 1425438846754932240)$
$( \, A_{25}, \; B_{25} \, ) \; = \; (5874690117631298042, \; 8308066439093374803)$

Note that

The sequence gives the values of the shortest leg of all Pythagorean triples   $(x, \; x+1, \; z)$
(3,   4,   5)
(20,   21,   29)
(119,   120,   169)
(696,   697,   985)
(4059,   4060,   5741)

and so on.

Also,

$B_{n} \cdot A_{n+1} \; = \; A_{n} \cdot B_{n+1} \; + \; A_{n}$

$B_1 A_2 \; = \; A_1 B_2 \; + \; A_1 \; = \; (3)(14) = (2)(20) + 2$
$B_2 A_3 \; = \; A_2 B_3 \; + \; A_2 \; = \; (20)(84) = (14)(119) + 14$
$B_3 A_4 \; = \; A_3 B_4 \; + \; A_3 \; = \; (119)(492) = (84)(696) + 84$

and so on