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
KEYWORD
nonn
AUTHOR
Eric M. Schmidt, Jan 27 2013
STATUS
approved