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

1,8

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

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.

Formula

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

Example

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

Prog

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

Crossrefs

Cf. A046079, A221838.

Sequence in context: A373431 A174126 A075613 * A189150 A155194 A080044

Adjacent sequences: A221834 A221835 A221836 * A221838 A221839 A221840

Keyword

nonn

Author

Eric M. Schmidt, Jan 27 2013

Status

approved