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%I #45 Jul 23 2023 01:52:43
%S 2,6,12,20,24,30,34,36,42,52,56,66,70,72,74,88,90,96,102,108,110,126,
%T 132,138,142,156,160,162,182,186,192,196,198,204,210,214,222,228,234,
%U 236,240,264,272,294,300,306,312,318,322,324,330,342,344,354,360,366,376,380,384,394
%N Integers k such that k^2 is of form y^2 + y + x^2 + x for positive y, x.
%C Integers k such that k^2 is the sum of two oblong numbers.
%C All terms are even. First k^2 that are expressible in m>1 ways:
%C m = 2, k = 12, {x,y} = {3,11},{8,8};
%C m = 3, k = 1168, {x,y} = {163,1156},{508,1051},{688,943};
%C m = 4, k = 600, {x,y} = {24,599},{135,584},{155,579},{260,540};
%C m = 5, k = 1746, {x,y} = {110,1742},{285,1722},{722,1589},{917,1485},{1062,1385};
%C m = 6, k = 6370, {x,y} = {1009,6289},{1330,6229},{2365,5914},{2665,5785},{3850,5074},{4105,4870}};
%C m = 8, k = 7332, {x,y} = {476,7316},{1083,7251},{1443,7188},{2036,7043},{3863,6231},{4368,5888},{4656,5663},{5111,5256}};
%C m = 9, k = 1734590;
%C m = 10, k = 1501632;
%C m = 12, k = 53766, {x,y} = {3537,53649},{6774,53337},{7625,53222},{8325,53117},{18317,50549},{19122,50250},{22125,49002},{22677,48749},{31077,43874},{32342,42950},{32825,42582},{35982,39950};
%C m = 13, k = 15994428;
%C m = 14, k = 36583944;
%C m = 16, k = 68906, {x,y} = {262,68905},{694,68902},{2242,68869},{14869,67282},{15802,67069},{16117,66994},{17305,66697},{26569,63577},{30430,61822},{30817,61630},{32194,60922},{33037,60469},{39877,56194},{43594,53362},{44782,52369},{45505,51742};
%C m = 18, k = 795918;
%C m = 20, k = 1501632;
%C m = 24, k = 338142;
%C m = 27, k = 216000900;
%C m = 32, k = 12464536;
%C m = 36, k = 6499622;
%C m = 40, k = 121608322;
%C m = 48, k = 10922046;
%C m = 64, k = 4210146;
%C m = 72, k = 207256338;
%C m = 96, k = 162026706;
%C Note that for k up 10^5 cases m = 7, 9, 10, 11, 13-15 are absent.
%C In general, odd values of m are much rarer than even values, why?
%C Case m = 7 is absent for k < 2*10^6. - _Giovanni Resta_, Jan 17 2014
%C Case m = 7 is absent for k <= 4*10^8. - _Zak Seidov_, Jan 26 2014
%H Zak Seidov, <a href="/A235375/b235375.txt">Table of n, a(n) for n = 1..11196</a> (all terms up to 10^5)
%Y Cf. A000290 (squares), A002378 (oblong numbers).
%K nonn
%O 1,1
%A _Zak Seidov_, Jan 17 2014
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