## When the Concatenation of n || (n+1) is Octagonal

Octagonal numbers are of the form    $O_{n} \; = \; n \, (3 \, n - 2)$

We can write

$10^{n} \; x \; + \; (x + 1) \; = \; y \, (3 \, y - 2)$

Or

$(10^{n} + 1) \, x \; + \; 1 \; = \; y \, (3 \, y - 2)$

the integer length of   $x$   is   $n$

Here are the first few solutions

n   =   1,   2,   5,   10 :     no solutions

n = 3

$100 \; || \; 101 \; = \; 100101 \; = \; O_{183}$
$704 \; || \; 705 \; = \; 704705 \; = \; O_{485}$

n = 4

$8132 \; || \; 8133 \; = \; 81328133 \; = \; O_{5207}$

n = 6

$142340 \; || \; 142341 \; = \; 142340142341 \; = \; O_{217823}$
$604384 \; || \; 604385 \; = \; 604384604385 \; = \; O_{448845}$

n = 7

$2754820 \; || \; 2754821 \; = \; 27548202754821 \; = \; O_{3030303}$
$3966944 \; || \; 3966945 \; = \; 39669443966945 \; = \; O_{3636365}$

n = 8

$29527104 \; || \; 29527105 \; = \; 2952710429527105 \; = \; O_{31372549}$
$37370244 \; || \; 37370245 \; = \; 3737024437370245 \; = \; O_{35294119}$

n = 9

$123636159 \; || \; 123636160 \; = \; 123636159123636160 \; = \; O_{203007520}$
$147556740 \; || \; 147556741 \; = \; 147556740147556741 \; = \; O_{221778223}$
$238451204 \; || \; 238451205 \; = \; 238451204238451205 \; = \; O_{281928599}$
$267667399 \; || \; 267667400 \; = \; 267667399267667400 \; = \; O_{298701300}$
$284250207 \; || \; 284250208 \; = \; 284250207284250208 \; = \; O_{307814992}$
$386323575 \; || \; 386323576 \; = \; 386323575386323576 \; = \; O_{358851676}$
$406195535 \; || \; 406195536 \; = \; 406195535406195536 \; = \; O_{367965368}$
$444070144 \; || \; 444070145 \; = \; 444070144444070145 \; = \; O_{384738069}$
$593777184 \; || \; 593777185 \; = \; 593777184593777185 \; = \; O_{444888445}$
$644939415 \; || \; 644939416 \; = \; 644939415644939416 \; = \; O_{463659148}$
$733607055 \; || \; 733607056 \; = \; 733607055733607056 \; = \; O_{494505496}$
$876687424 \; || \; 876687425 \; = \; 876687424876687425 \; = \; O_{540582225}$
$922929407 \; || \; 922929408 \; = \; 922929407922929408 \; = \; O_{554655872}$
$979591840 \; || \; 979591841 \; = \; 979591840979591841 \; = \; O_{571428573}$

$O_{485} \; = \; 704705 \; = \; 704 \; || \; 705$
$O_{448845} \; = \; 604384604385 \; = \; 604384 \; || \; 604385$
$O_{444888445} \; = \; 593777184593777185 \; = \; 593777184 \; || \; 593777185$
$O_{444488884445} \; = \; 592711105184592711105185 \; = \; 592711105184 \; || \; 592711105185$

the pattern continues