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

1,1

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

Table of n, a(n) for n=1..47.

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

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

Subsequence of A188158.

Sequence in context: A230491 A067249 A351980 * A350947 A288321 A155191

Adjacent sequences: A351175 A351176 A351177 * A351179 A351180 A351181

Keyword

nonn

Author

Bernard Schott, Feb 04 2022

Extensions

Missing terms added by Michel Marcus, Feb 05 2022

Status

approved