## Triangular Num3ers : x^2 + y^2 – x = 6*x*y

Consider the equation      $x^2 \; + \; y^2 \; - \; x \; = \; 6 \, x \, y$

Here are the first few solutions:

$x = 1 = 1^2$
y = 0,    y = 6
$x = 36 = 6^2$
y = 6,    y = 210
$x = 1225 = 35^2$
y = 210,    y = 7140
$x = 41616 = 204^2$
y = 7140,    y = 242556
$x = 1413721 = 1189^2$
y = 242556,    y = 8239770
$x = 48024900 = 6930^2$
y = 8239770,    y = 279909630
$x = 1631432881 = 40391^2$
y = 279909630,    y = 9508687656
$x = 55420693056 = 235416^2$
y = 9508687656,    y = 323015470680
$x = 1882672131025 = 1372105^2$
y = 323015470680,    y = 10973017315470
$x = 63955431761796 = 7997214^2$
y = 10973017315470,    y = 372759573255306
$x = 2172602007770041 = 46611179^2$
y = 372759573255306,    y = 12662852473364940
$x = 73804512832419600 = 271669860^2$
y = 12662852473364940,    y = 430164224521152660
$x = 2507180834294496361 = 1583407981^2$
y = 430164224521152660,    y = 14612920781245825506
$x = 85170343853180456676 = 9228778026^2$
y = 14612920781245825506,    y = 496409142337836914550
$x = 2893284510173841030625 = 53789260175^2$
y = 496409142337836914550,    y = 16863297918705209269200

$x = 1 = 1^2 = T_1$
$x = 36 = 6^2 = T_{8} = T_{3^2 - 1}$
$x = 1225 = 35^2 = T_{49} = T_{7^2}$
$x = 41616 = 204^2 = T_{288} = T_{17^2 - 1}$
$x = 1413721 = 1189^2 = T_{1681} = T_{41^2}$
$x = 48024900 = 6930^2 = T_{9800} = T_{99^2 - 1}$
$x = 1631432881 = 40391^2 = T_{57121} = T_{239^2}$
$x = 55420693056 = 235416^2 = T_{332928} = T_{577^2 - 1}$
$x = 1882672131025 = 1372105^2 = T_{1940449} = T_{1393^2}$
$x = 63955431761796 = 7997214^2 = T_{11309768} = T_{3363^2 - 1}$
$x = 2172602007770041 = 46611179^2 = T_{65918161} = T_{8119^2}$
$x = 73804512832419600 = 271669860^2 = T_{384199200} = T_{19601^2 - 1}$
$x = 2507180834294496361 = 1583407981^2 = T_{2239277041} = T_{47321^2}$
$x = 85170343853180456676 = 9228778026^2 = T_{13051463048} = T_{114243^2 - 1}$
$x = 2893284510173841030625 = 53789260175^2 = T_{76069501249} = T_{275807^2}$

$8(x) + 1$    is a square

$8(1) + 1 = 3^2$
$T_{3^2 - 1} = 6^2$
$8(6^2) + 1 = 17^2$
$T_{17^2 - 1} = 204^2$
$8(35^2) + 1 = 99^2$
$T_{99^2 - 1} = 6930^2$
$8(204^2) + 1 = 577^2$
$T_{577^2 - 1} = 235416^2$
$8(1189^2) + 1 = 3363^2$
$T_{3363^2 - 1} = 7997214^2$
$8(6930^2) + 1 = 19601^2$
$T_{19601^2 - 1} = 271669860^2$
$8(40391^2) + 1 = 114243^2$
$T_{114243^2 - 1} = 9228778026^2$
$8(235416^2) + 1 = 665857^2$
$T_{665857^2 - 1} = 313506783024^2$
$8(1372105^2) + 1 = 3880899^2$
$T_{3880899^2 - 1} = 10650001844790^2$
$8(7997214^2) + 1 = 22619537^2$
$T_{22619537^2 - 1} = 361786555939836^2$
$8(46611179^2) + 1 = 131836323^2$
$T_{131836323^2 - 1} = 12290092900109634^2$
$8(271669860^2) + 1 = 768398401^2$
$T_{768398401^2 - 1} = 417501372047787720^2$
$8(1583407981^2) + 1 = 4478554083^2$
$T_{4478554083^2 - 1} = 14182756556724672846^2$
$8(9228778026^2) + 1 = 26102926097^2$
$T_{26102926097^2 - 1} = 481796221556591089044^2$
$8(53789260175^2) + 1 = 152139002499^2$
$T_{152139002499^2 - 1} = 16366888776367372354650^2$

Using the square roots:

$(6+n) \,(6+n+1)/2 \; = \; n \,(n+1)$ …………………. n = 14
$T_{20} \; = \; 2 \: T_{14}$

$(35+n) \,(35+n+1)/2 \; = \; n \,(n+1)$ ……………….. n = 84
$T_{119} \; = \; 2 \: T_{84}$

$(204+n) \,(204+n+1)/2 \; = \; n \,(n+1)$ ……………… n = 492
$T_{696} \; = \; 2 \: T_{492}$

$(1189+n) \,(1189+n+1)/2 \; = \; n \,(n+1)$ ……………. n = 2870
$T_{4059} \; = \; 2 \: T_{2870}$

$(6930+n) \,(6930+n+1)/2 \; = \; n \,(n+1)$ ……………. n = 16730
$T_{23660} \; = \; 2 \: T_{16730}$

$(40391+n) \,(40391+n+1)/2 \; = \; n \,(n+1)$ ………….. n = 97512
$T_{137903} \; = \; 2 \: T_{97512}$

$(235416+n) \,(235416+n+1)/2 \; = \; n \,(n+1)$ ………… n = 568344
$T_{803760} \; = \; 2 \: T_{568344}$

$(1372105+n) \,(1372105+n+1)/2 \; = \; n \,(n+1)$ ………. n = 3312554
$T_{4684659} \; = \; 2 \: T_{3312554}$

$(7997214+n) \,(7997214+n+1)/2 \; = \; n \,(n+1)$ ………. n = 19306982
$T_{27304196} \; = \; 2 \: T_{19306982}$

$(46611179+n) \,(46611179+n+1)/2 \; = \; n \,(n+1)$ …….. n = 112529340
$T_{159140519} \; = \; 2 \: T_{112529340}$

$(271669860+n) \,(271669860+n+1)/2 \; = \; n \,(n+1)$ …… n = 655869060
$T_{927538920} \; = \; 2 \: T_{655869060}$

$(1583407981+n) \,(1583407981+n+1)/2 \; = \; n \,(n+1)$ …. n = 3822685022
$T_{5406093003} \; = \; 2 \: T_{3822685022}$

$(9228778026+n) \,(9228778026+n+1)/2 \; = \; n \,(n+1)$ …. n = 22280241074
$T_{31509019100} \; = \; 2 \: T_{22280241074}$

$(53789260175+n) \,(53789260175+n+1)/2 \; = \; n \,(n+1)$ … n = 129858761424
$T_{183648021599} \; = \; 2 \: T_{129858761424}$