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

%I A221837 #14 Feb 16 2025 08:33:19
%S A221837 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,
%T A221837 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,
%U A221837 1,1,169,1,1,49,25,16,16,1,16,16,16,1,144,1,1,49
%N A221837 Number of integer Heron triangles of height n such that the angles adjacent to the base are not right.
%C A221837 The number of integer Heron triangles of height n with a right angle at the base is given by A046079.
%C A221837 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
%H A221837 Eric M. Schmidt, <a href="/A221837/b221837.txt">Table of n, a(n) for n = 1..10000</a>
%H A221837 Sourav Sen Gupta, Nirupam Kar, Subhamoy Maitra, Santanu Sarkar, and Pantelimon Stanica, <a href="http://www.emis.de/journals/INTEGERS/papers/n3/n3.Abstract.html">Counting Heron triangles with Constraints</a>, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A3, 2013.
%H A221837 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeronianTriangle.html">Heronian Triangle.</a>
%F A221837 a(n) = A046079(n)^2.
%e A221837 The triangle for n = 3 has side lengths (5, 5, 8) and area 12.
%o A221837 (Sage) def A221837(n) : return (number_of_divisors(n^2 if n%2==1 else (n/2)^2) - 1)^2 // 4
%Y A221837 Cf. A046079, A221838.
%K A221837 nonn
%O A221837 1,8
%A A221837 _Eric M. Schmidt_, Jan 27 2013