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A351178 Integral area of primitive integer-sided triangles whose sides a < b < c are in arithmetic progression.

%I A351178 #45 Feb 06 2022 21:07:44
%S A351178 6,84,126,156,210,456,546,570,1116,1170,1176,1554,2046,2220,2394,3096,
%T A351178 3216,3294,3354,3924,4740,5124,6006,6180,6510,7326,7446,8760,9030,
%U A351178 9264,9906,10374,10920,11466,12684,13104,15210,16170,16296,16716,17556,18060,18090,18354,22134,22860,23550
%N A351178 Integral area of primitive integer-sided triangles whose sides a < b < c are in arithmetic progression.
%C A351178 Middle side b is necessarily even, and the two other sides are odd, so all the areas are even numbers.
%C A351178 If b is the middle side with b even >= 4, if k odd = b-a = c-b with 1 <= k <= b/2 - 1, if gcd(b,k) = 1, then, we have area S = sqrt(3*b^2*(b^2-4*k^2))/4.
%D A351178 David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, entry 126, page 122.
%H A351178 John MacNeill, <a href="https://drive.google.com/file/d/1oBW1NrZlzGPqKWTNHUzjnnsIx3pIi6et/view">13, 14, 15 and 15, 26, 37</a>, Mathematical Spectrum, Vol. 21, No. 3, 1989, pp. 83-84.
%e A351178 a(1) = 6 corresponds to the Pythagorean triple (3, 4, 5), this is the unique right integer-sided triangle in this sequence.
%e A351178 a(2) = 84 for triple (13, 14, 15) (see MacNeill link).
%e A351178 a(3) = 126 for triple (15, 28, 41) (see Penguin reference, entry 126).
%e A351178 a(4) = 156 for triple (15, 26, 37) (see MacNeill link).
%o A351178 (PARI) lista(nn) = {my(list = List()); for (b = 3, nn, for (a = b-floor((b-1)/2), b-1, my(c = 2*b - a); if (gcd([a,b,c]) == 1, my(p = (a+b+c)/2); if (issquare(x=p*(p-a)*(p-b)*(p-c)), listput(list, sqrtint(x)));););); vecsort(Vec(list));} \\ _Michel Marcus_, Feb 05 2022
%Y A351178 Cf. A336750, A336755, A336750, A336756, A336757.
%Y A351178 Subsequence of A188158.
%K A351178 nonn
%O A351178 1,1
%A A351178 _Bernard Schott_, Feb 04 2022
%E A351178 Missing terms added by _Michel Marcus_, Feb 05 2022