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%I #7 Mar 30 2012 18:34:45
%S 3,102,130,312,759,2496,2706,3465,6072,6111,8424,14004,16005,36897,
%T 37156,92385,98640,112032,117708,128040,351260,378108,740050,1346400,
%U 1371900,1898130,3998607,5986575,6082065,6631596,6741214,7692804
%N Numbers whose square can be expressed as a+b*c, with a,b,c in geometric sequence.
%C This sequence was inspired by a puzzle question that asked for all squares under a trillion that can be expressed as either a+bc or b+ac where a,b,c are in increasing geometric progression.
%e a(1)=3 because 3^2=1+2*4 and 1,2,4 are in geometric sequence
%e a(2)=102 because 102^2=36+72*144 and 36,72,144 are in geometric sequence
%e a(3)=130 because 130^2=25+75*225 and 25,75,225 are in geometric sequence
%e a(4)=312 because 312^2=8+92*1058 and ...
%e a(5)=759 because 759^2=81+360*1600
%e a(6)=2496 because 2496^2=512+1472*4232
%e a(7)=2706 because 2706^2=1936+2420*3025
%e a(8)=3465 because 3465^2=1225+2450*4900
%e a(9)=6072 because 6072^2=5184+5760*6400
%e a(10)=6111 because 6111^2=3969+5292*7056
%e a(11)=8424 because 8424^2=5832+7452*9522
%e a(12)=14004 because 14004^2=432+4392*44652
%e a(13)=16005 because 16005^2=1089+6534*39204
%e a(14)=36897 because 36897^2=21609+30870*44100
%e a(15)=37156 because 37156^2=12544+25872*53361
%e a(16)=92385 because 92385^2=50625+75600*112896
%e a(17)=98640 because 98640^2=50625+78975*123201
%e a(18)=112032 because 112032^2=27648+70272*178608
%e a(19)=117708 because 117708^2=41616+83232*166464
%e a(20)=128040 because 128040^2=69696+104544*156816
%e a(21)=351260 because 351260^2=67600+202800*608400
%e a(22)=378108 because 378108^2=314928+355752*401868
%e a(23)=740050 because 740050^2=521284+658464*831744
%e No other numbers smaller than a million have squares that can be expressed this way.
%e Contribution from _Donovan Johnson_, Jul 30 2010: (Start)
%e a(24)=1346400 because 1346400^2=135000+625500*2898150
%e a(25)=1371900 because 1371900^2=10000+266000*7075600
%e a(26)=1898130 because 1898130^2=6084+279864*12873744
%e a(27)=3998607 because 3998607^2=1413721+2827442*5654884
%e a(28)=5986575 because 5986575^2=1157625+3461850*10352580
%e a(29)=6082065 because 6082065^2=4348377+5438466*6801828
%e a(30)=6631596 because 6631596^2=1944+440532*99829446
%e a(31)=6741214 because 6741214^2=334084+2476152*18352656
%e a(32)=7692804 because 7692804^2=444528+2974104*19898172
%e (End)
%K nonn
%O 1,1
%A _Graeme McRae_, Jul 05 2007
%E Added word: 'increasing'. The original puzzle was expressed as a modulo operation, the expression was 'remainder + quotient * divisor', where the remainder is necessarily smaller than the divisor, implying an increasing sequence. Counterexample if 'increasing' is not specified: a=8, b=4, c=2. a+b*c = 16 = 4^2; 4 is not in sequence A130733 - James Cunnane (james.cunnane(AT)gmail.com), Jun 29 2010
%E a(24)-a(32) from _Donovan Johnson_, Jul 30 2010
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