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A306212 Numbers that are the sum of squares of three distinct positive integers in arithmetic progression.

%I A306212 #18 Jun 09 2020 03:42:17
%S A306212 14,29,35,50,56,66,77,83,93,107,110,116,126,140,149,155,158,165,179,
%T A306212 194,197,200,210,219,224,242,245,251,261,264,275,290,293,302,308,315,
%U A306212 318,332,341,350,365,371,372,381,395,398,413,428,434,435,440,450,461,462,464,482
%N A306212 Numbers that are the sum of squares of three distinct positive integers in arithmetic progression.
%H A306212 Robert Israel, <a href="/A306212/b306212.txt">Table of n, a(n) for n = 1..10000</a>
%e A306212 35 = 1^2 + 3^2 + 5^2, with 3 - 1 = 5 - 3 = 2;
%e A306212 371 = 1^2 + 9^2 + 17^2, with 9 - 1 = 17 - 9 = 8. Also 371 = 9^2 + 11^2 + 13^2, with 11 - 9 = 13 - 11 = 2.
%p A306212 N:= 1000: # for terms <= N
%p A306212 S:= {seq(seq(3*a^2+2*b^2, b=1..min(a-1, floor(sqrt((N-3*a^2)/2)))),a=1..floor(sqrt(N/3)))}:
%p A306212 sort(convert(S,list)); # _Robert Israel_, Jun 08 2020
%o A306212 (PARI) for(n=3, 600, k=sqrt(n/3); a=2; v=0; while(a<=k&&v==0, b=(n-3*a^2)/2; if(b==truncate(b)&&issquare(b), d=sqrt(b); if(d>=1&&d<=a-1, v=1; print1(n,", "))); a+=1))
%o A306212 (PARI) w=List(); for(n=3, 600, k=sqrt(n/3); for(a=2, k, for(c=1, a-1, v=(a-c)^2+a^2+(a+c)^2; if(v==n, listput(w,n))))); print(vecsort(Vec(w),,8))
%Y A306212 Cf. A000290, A000378, A000408, A085317, A120328, A292313, A306213, A306214.
%K A306212 nonn
%O A306212 1,1
%A A306212 _Antonio Roldán_, Jan 29 2019