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

6, 84, 126, 156, 210, 456, 546, 570, 1116, 1170, 1176, 1554, 2046, 2220, 2394, 3096, 3216, 3294, 3354, 3924, 4740, 5124, 6006, 6180, 6510, 7326, 7446, 8760, 9030, 9264, 9906, 10374, 10920, 11466, 12684, 13104, 15210, 16170, 16296, 16716, 17556, 18060, 18090, 18354, 22134, 22860, 23550

Offset: 1

Author

Bernard Schott, Feb 04 2022

Keywords

• nonn

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.

Examples

			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).
		

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.

Crossrefs

Cf. A336750, A336755, A336750, A336756, A336757.

Subsequence of A188158.

Programs

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

Extensions

Missing terms added by Michel Marcus, Feb 05 2022