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

Heptagonal numbers are of the form    $n \,(5 \, n - 3)/2$

We can write

$10^{n} \; x \; + \; (x + 1) \; = \; y \,(5 \, y - 3)/2$

Or

$(10^{n} + 1) \, x \; + \; 1 \; = \; y \,(5 \, y - 3)/2$

the integer length of   $x$   is   $n$

Here are the first few solutions

n = 1

$3 \; || \; 4 \; = \; 34 \; = \; H_4$

n = 2

$39 \; || \; 40 \; = \; 3940 \; = \; H_{40}$

n = 3

$238 \; || \; 239 \; = \; 238239 \; = \; H_{309}$
$399 \; || \; 400 \; = \; 399400 \; = \; H_{400}$

n = 4

$1521 \; || \; 1522 \; = \; 15211522 \; = \; H_{2467}$
$3999 \; || \; 4000 \; = \; 39994000 \; = \; H_{4000}$

n = 5

$39999 \; || \; 40000 \; = \; 3999940000 \; = \; H_{40000}$
$51655 \; || \; 51656 \; = \; 5165551656 \; = \; H_{45456}$

n = 6

$183991 \; || \; 183992 \; = \; 183991183992 \; = \; H_{271287}$
$399999 \; || \; 400000 \; = \; 399999400000 \; = \; H_{400000}$

n = 7

$1190082 \; || \; 1190083 \; = \; 11900821190083 \; = \; H_{2181818}$
$3999999 \; || \; 4000000 \; = \; 39999994000000 \; = \; H_{4000000}$

n = 8

$39999999 \; || \; 40000000 \; = \; 3999999940000000 \; = \; H_{40000000}$
$55363324 \; || \; 55363325 \; = \; 5536332455363325 \; = \; H_{47058825}$

n = 9

$173130195 \; || \; 173130196 \; = \; 173130195173130196 \; = \; H_{263157896}$
$178399921 \; || \; 178399922 \; = \; 178399921178399922 \; = \; H_{267132867}$
$330578514 \; || \; 330578515 \; = \; 330578514330578515 \; = \; H_{363636365}$
$337845235 \; || \; 337845236 \; = \; 337845235337845236 \; = \; H_{367611336}$
$392089558 \; || \; 392089559 \; = \; 392089558392089559 \; = \; H_{396025029}$
$399999999 \; || \; 400000000 \; = \; 399999999400000000 \; = \; H_{400000000}$
$982177609 \; || \; 982177610 \; = \; 982177609982177610 \; = \; H_{626794260}$
$994674555 \; || \; 994674556 \; = \; 994674555994674556 \; = \; H_{630769231}$

n = 10

$1921380258 \; || \; 1921380259 \; = \; 19213802581921380259 \; = \; H_{2772277229}$
$2861552983 \; || \; 2861552984 \; = \; 28615529832861552984 \; = \; H_{3383225079}$
$3999999999 \; || \; 4000000000 \; = \; 39999999994000000000 \; = \; H_{4000000000}$
$9472552161 \; || \; 9472552162 \; = \; 94725521619472552162 \; = \; H_{6155502307}$