%I #34 Jul 05 2022 07:16:21

%S 8,15,20,24,27,32,35,39,44,48,49,51,54,55,56,63,64,65,68,75,80,84,87,

%T 90,92,95,98,99,104,111,114,116,119,120,123,125,128,132,135,140,143,

%U 144,147,152,153,155,159,160,164,168,170,171,174,175,176,183,184,185,188,189,195,200,203,204,207,208,209,212,215,216,219,220,224,230,231

%N Numbers of the form a+b+c = a^2 - b^2 - c^2 where a > b >= c > 0.

%C It seems that A082867 is a subsequence.

%C The first counterexample to the above is A082867(60) = 258. - _Charles R Greathouse IV_, Jul 05 2022

%H Charles R Greathouse IV, <a href="/A355490/b355490.txt">Table of n, a(n) for n = 1..10000</a>

%e 8 is a term: 8 = 4+2+2 = 4^2 - 2^2 - 2^2.

%e 15 is a term: 15 = 7+5+3 = 7^2 - 5^2 - 3^2.

%t Solve[a==r^2-s^2-d^2 && 1<=r<=120 && 1<=s<=120 && 1<=d<=120 && 0<=a && r>s>=d && a==r+s+d, {a,r,s,d}, Integers]

%o (PARI) list(lim)=my(v=List([8]));lim\=1;for(a=3,lim-2,my(a2=a^2);for(b=(sqrt(2*a^2+2*a+1)-1)\2,a-2,my(t=a2-b^2-a-b,s);if(issquare(4*t+1,&s) && (c=(s-1)/2)<=b && c<=b && a+b+c<=lim, listput(v,a+b+c)))); Set(v) \\ _Charles R Greathouse IV_, Jul 05 2022

%Y Cf. A082867, A082772, A134582, A355491.

%K nonn

%O 1,1

%A _Mohammad Arab_, Jul 04 2022

%E a(57) = 184 inserted by _Charles R Greathouse IV_, Jul 05 2022