A221837 Number of integer Heron triangles of height n such that the angles adjacent to the base are not right.

0, 0, 1, 1, 1, 1, 1, 4, 4, 1, 1, 16, 1, 1, 16, 9, 1, 4, 1, 16, 16, 1, 1, 49, 4, 1, 9, 16, 1, 16, 1, 16, 16, 1, 16, 49, 1, 1, 16, 49, 1, 16, 1, 16, 49, 1, 1, 100, 4, 4, 16, 16, 1, 9, 16, 49, 16, 1, 1, 169, 1, 1, 49, 25, 16, 16, 1, 16, 16, 16, 1, 144, 1, 1, 49

Offset: 1

Author

Eric M. Schmidt, Jan 27 2013

Keywords

• nonn

Comments

The number of integer Heron triangles of height n with a right angle at the base is given by A046079.

a(n) is the number of distinct triangles with height n that can be formed with two right triangles with integer sides, either by joining them or by cutting off the smaller one from the larger one. In both cases, the two right triangles must have a leg of length n. To form a triangle with height n, there are binomial(A046079(n) + 1, 2) ways to join them and binomial(A046079(n), 2) ways to cut off the smaller one from the larger one. That's a total of (A046079(n)+1, 2) + (A046079(n), 2)= (A046079(n))^2. - Felix Huber, Aug 20 2023

Examples

			The triangle for n = 3 has side lengths (5, 5, 8) and area 12.
		

Links

• Eric M. Schmidt, Table of n, a(n) for n = 1..10000
• Sourav Sen Gupta, Nirupam Kar, Subhamoy Maitra, Santanu Sarkar, and Pantelimon Stanica, Counting Heron triangles with Constraints, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A3, 2013.
• Eric Weisstein's World of Mathematics, Heronian Triangle.

Crossrefs

Cf. A046079, A221838.

Programs

Sage

def A221837(n) : return (number_of_divisors(n^2 if n%2==1 else (n/2)^2) - 1)^2 // 4

Formula

a(n) = A046079(n)^2.