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

Hexagonal numbers are of the form    $n \, (2 \, n - 1)$

We can write

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

Or

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

the integer length of   $x$   is   $n$

Here are the first few solutions

n = 1

$4 \, || \, 5 \, = \, 45 \, = \, H_5$

n = 2

$49 \, || \, 50 \, = \, 4950 \, = \, H_{50}$

n = 3

$369 \, || \, 370 \, = \, 369370 \, = \, H_{430}$
$415 \, || \, 416 \, = \, 415416 \, = \, H_{456}$
$499 \, || \, 500 \, = \, 499500 \, = \, H_{500}$
$760 \, || \, 761 \, = \, 760761 \, = \, H_{617}$

n = 4

$4999 \, || \, 5000 \, = \, 49995000 \, = \, H_{5000}$
$8827 \, || \, 8828 \, = \, 88278828 \, = \, H_{6644}$

n = 5

$10330 \, || \, 10331 \, = \, 1033010331 \, = \, H_{22727}$
$14877 \, || \, 14878 \, = \, 1487714878 \, = \, H_{27274}$
$49999 \, || \, 50000 \, = \, 4999950000 \, = \, H_{50000}$

n = 6

$499999 \, || \, 500000 \, = \, 499999500000 \, = \, H_{500000}$
$490150 \, || \, 490151 \, = \, 490150490151 \, = \, H_{495051}$

n = 7

$4999999 \, || \, 5000000 \, = \, 49999995000000 \, = \, H_{5000000}$
$6983470 \, || \, 6983471 \, = \, 69834706983471 \, = \, H_{5909091}$

n = 8

$49999999 \, || \, 50000000 = 4999999950000000 \, = \, H_{50000000}$
$83737027 \, || \, 83737028 = 8373702783737028 \, = \, H_{64705884}$

n = 9

$167590027 \, || \, 167590028 \; = \; 167590027167590028 \; = \; H_{289473684}$
$215738734 \, || \, 215738735 \; = \; 215738734215738735 \; = \; H_{328434723}$
$249759739 \, || \, 249759740 \; = \; 249759739249759740 \; = \; H_{353383460}$
$254388969 \, || \, 254388970 \; = \; 254388969254388970 \; = \; H_{356643358}$
$255102040 \, || \, 255102041 \; = \; 255102040255102041 \; = \; H_{357142857}$
$444232045 \, || \, 444232046 \; = \; 444232045444232046 \; = \; H_{471291866}$
$499001500 \, || \, 499001501 \; = \; 499001500499001501 \; = \; H_{499500501}$
$499999999 \, || \, 500000000 \; = \; 499999999500000000 \; = \; H_{500000000}$
$579881659 \, || \, 579881660 \; = \; 579881659579881660 \; = \; H_{538461540}$
$580958002 \, || \, 580958003 \; = \; 580958002580958003 \; = \; H_{538961039}$
$643362877 \, || \, 643362878 \; = \; 643362877643362878 \; = \; H_{567169674}$
$739705014 \, || \, 739705015 \; = \; 739705014739705015 \; = \; H_{608155003}$
$928390294 \, || \, 928390295 \; = \; 928390294928390295 \; = \; H_{681318683}$
$929752065 \, || \, 929752066 \; = \; 929752065929752066 \; = \; H_{681818182}$

n = 10

$1066727340 \, || \, 1066727341 \; = \; 10667273401066727341 \; = \; H_{2309466757}$
$1447793827 \, || \, 1447793828 \; = \; 14477938271447793828 \; = \; H_{2690533244}$
$1764532890 \, || \, 1764532891 \; = \; 17645328901764532891 \; = \; H_{2970297031}$
$4999999999 \, || \, 5000000000 \; = \; 49999999995000000000 \; = \; H_{5000000000}$
$6408999877 \, || \, 6408999878 \; = \; 64089998776408999878 \; = \; H_{5660830274}$