## Concatenations : x || y = x^2/2 + 2*y^2

$x \; || \; y \; = \; x^2/2 \; + \; 2 \,(y^2)$

where   $x$   and   $y$   are positive integers

$12 || 24 \; = \; 1224 \; = 12^2/2 \; + \; 2(24^2)$
$188 || 24 \; = \; 18824 \; = \; 188^2/2 \; + \; 2(24^2)$

$12 \; + \; 188 \; = \; 2 \; \times \; 10^2$

$90 || 50 = 9050 = 90^2/2 + 2(50^2)$
$110 || 50 = 11050 = 110^2/2 + 2(50^2)$

$90 \; + \; 110 \; = \; 2 \; \times \; 10^2$

$780 || 488 = 780488 = 780^2/2 + 2(488^2)$
$1220 || 488 = 1220488 = 1220^2/2 + 2(488^2)$

$780 \; + \; 1220 \; = \; 2 \; \times \; 10^3$

$9900 || 5000 = 99005000 = 9900^2/2 + 2(5000^2)$
$10100 || 5000 = 101005000 = 10100^2/2 + 2(5000^2)$

$9900 \; + \; 10100 \; = \; 2 \; \times \; 10^4$

$61538 || 46154 = 6153846154 = 61538^2/2 + 2(46154^2)$
$138462 || 46154 = 13846246154 = 138462^2/2 + 2(46154^2)$

$61538 \; + \; 138462 \; = \; 2 \; \times \; 10^5$

$584838 || 454874 = 584838454874 = 584838^2/2 + 2(454874^2)$
$1415162 || 454874 = 1415162454874 = 1415162^2/2 + 2(454874^2)$

$584838 \; + \; 1415162 \; = \; 2 \; \times \; 10^6$

$999000 || 500000 = 999000500000 = 999000^2/2 + 2(500000^2)$
$1001000 || 500000 = 1001000500000 = 1001000^2/2 + 2(500000^2)$

$999000 \; + \; 1001000 \; = \; 2 \; \times \; 10^6$

$9174962 || 4982954 = 91749624982954 = 9174962^2/2 + 2(4982954^2)$
$10825038 || 4982954 = 108250384982954 = 10825038^2/2 + 2(4982954^2)$

$9174962 \; + \; 10825038 \; = \; 2 \; \times \; 10^7$

$27038 || 0 || 1162634 = 2703801162634 = 27038^2/2 + 2(1162634)^2$
$199972962 || 0 || 1162634 = 19997296201162634 = 199972962^2/2 + 2(1162634)^2$

$27038 \; + \; 199972962 \; = \; 2 \; \times \; 10^8$

$1500938 || 0 || 8630394 = 150093808630394 = 1500938^2/2 + 2(8630394^2)$
$198499062 || 0 || 8630394 = 19849906208630394 = 198499062^2/2 + 2(8630394^2)$

$1500938 \; + \; 198499062 \; = \; 2 \; \times \; 10^8$

$2441220 || 10980488 = 244122010980488 = 2441220^2/2 + 2(10980488^2)$
$197558780 || 10980488 = 19755878010980488 = 197558780^2/2 + 2(10980488^2)$

$2441220 \; + \; 197558780 \; = \; 2 \; \times \; 10^8$

$7002188 || 18380744 = 700218818380744 = 7002188^2/2 + 2(18380744^2)$
$192997812 || 18380744 = 19299781218380744 = 192997812^2/2 + 2(18380744^2)$

$7002188 \; + \; 192997812 \; = \; 2 \; \times \; 10^8$

$7193610 || 18621050 = 719361018621050 = 7193610^2/2 + 2(18621050^2)$
$192806390 || 18621050 = 19280639018621050 = 192806390^2/2 + 2(18621050^2$

$7193610 \; + \; 192806390 \; = \; 2 \; \times \; 10^8$

$7688462 || 19226154 = 768846219226154 = 7688462^2/2 + 2(19226154^2)$
$192311538 || 19226154 = 19231153819226154 = 192311538^2/2 + 2(19226154^2)$

$7688462 \; + \; 192311538 \; = \; 2 \; \times \; 10^8$

$14743350 || 26130938 = 1474335026130938 = 14743350^2/2 + 2(26130938^2)$
$185256650 || 26130938 = 18525665026130938 = 185256650^2/2 + 2(26130938^2)$

$14743350 \; + \; 185256650 \; = \; 2 \; \times \; 10^8$

$27382838 || 34375674 = 2738283834375674 = 27382838^2/2 + 2(34375674^2)$
$172617162 || 34375674 = 17261716234375674 = 172617162^2/2 + 2(34375674^2)$

$27382838 \; + \; 172617162 \; = \; 2 \; \times \; 10^8$

$28287930 || 34847450 = 2828793034847450 = 28287930^2/2 + 2(34847450^2)$
$171712070 || 34847450 = 17171207034847450 = 171712070^2/2 + 2(34847450^2)$

$28287930 \; + \; 171712070 \; = \; 2 \; \times \; 10^8$

$47746650 || 42630938 = 4774665042630938 = 47746650^2/2 + 2(42630938^2)$
$152253350 || 42630938 = 15225335042630938 = 152253350^2/2 + 2(42630938^2)$

$47746650 \; + \; 152253350 \; = \; 2 \; \times \; 10^8$

$63247812 || 46500744 = 6324781246500744 = 63247812^2/2 + 2(46500744^2)$
$136752188 || 46500744 = 13675218846500744 = 136752188^2/2 + 2(46500744^2)$

$63247812 \; + \; 136752188 \; = \; 2 \; \times \; 10^8$

$78048780 || 48780488 = 7804878048780488 = 78048780^2/2 + 2(48780488^2)$
$121951220 || 48780488 = 12195122048780488 = 121951220^2/2 + 2(48780488^2)$

$78048780 \; + \; 121951220 \; = \; 2 \; \times \; 10^8$

$82749062 || 49250394 = 8274906249250394 = 82749062^2/2 + 2(49250394^2)$
$117250938 || 49250394 = 11725093849250394 = 117250938^2/2 + 2(49250394^2)$

$82749062 \; + \; 117250938 \; = \; 2 \; \times \; 10^8$

$84549288 || 49399584 = 8454928849399584 = 84549288^2/2 + 2(49399584^2)$
$115450712 || 49399584 = 11545071249399584 = 115450712^2/2 + 2(49399584^2)$

$84549288 \; + \; 115450712 \; = \; 2 \; \times \; 10^8$

$99990000 || 50000000 = 9999000050000000 = 99990000^2/2 + 2(50000000^2)$
$100010000 || 50000000 = 10001000050000000 = 100010000^2/2 + 2(50000000^2)$

$99990000 \; + \; 100010000 \; = \; 2 \; \times \; 10^8$

$924704538 || 498580634 = 924704538498580634 = 924704538^2/2 + 2(498580634)^2$
$1075295462 || 498580634 = 1075295462498580634 = 1075295462^2/2 + 2(498580634)^2$

$924704538 \; + \; 1075295462 \; = \; 2 \; \times \; 10^9$