OFFSET
1,1
COMMENTS
Middle side b is necessarily even, and the two other sides are odd, so all the areas are even numbers.
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.
REFERENCES
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, entry 126, page 122.
LINKS
John MacNeill, 13, 14, 15 and 15, 26, 37, Mathematical Spectrum, Vol. 21, No. 3, 1989, pp. 83-84.
EXAMPLE
a(1) = 6 corresponds to the Pythagorean triple (3, 4, 5), this is the unique right integer-sided triangle in this sequence.
a(2) = 84 for triple (13, 14, 15) (see MacNeill link).
a(3) = 126 for triple (15, 28, 41) (see Penguin reference, entry 126).
a(4) = 156 for triple (15, 26, 37) (see MacNeill link).
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Bernard Schott, Feb 04 2022
EXTENSIONS
Missing terms added by Michel Marcus, Feb 05 2022
STATUS
approved
