A221838 -id:A221838 - cp's OEIS frontend

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

Eric M. Schmidt, Jan 27 2013

nonn

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

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

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.

Cf. A046079, A221838.

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

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

A370599 a(n) is the number of distinct triangles with integral side-lengths for which the perimeter 2*n divides the area.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 1, 1, 2, 2, 0, 0, 1, 1, 0, 4, 0, 1, 3, 0, 0, 1, 0, 0, 0, 0, 0, 3, 1, 2, 1, 0, 0, 3, 0, 0, 3, 2, 1, 4, 0, 2, 0, 3, 0, 2, 0, 0, 2, 2, 3, 1, 0, 2, 4, 1, 0, 8, 1, 0, 1

Offset: 1

Felix Huber, Mar 09 2024

If the perimeter 2*n of a triangle with integral edge-lengths divides its area A, then this also applies to a triangle stretched with positive integer k, because A*k^2/(2*n*k) = k*A/(2*n). Therefore a(d) <= a(n) for all positive divisors d of n and a(m) >= a(n) for all positive integer multiples m of n.

With an odd perimeter, according to Heron's formula the area A would have the form A = sqrt((2*k - 1)/8), where k is a positive integer. The area A would be irrational and the integer perimeter would not divide the area A. For this reason, only triangles with an even perimeter are considered in this sequence.

			a(18) = 1, because only the triangle (9, 10, 17) satisfies the condition: A/(2*n) = 36/36 = 1. (9, 10, 17) is one of the five triangles for which the perimeter is equal to the area (see A098030).
a(42) = 4, because exactly the 4 triangles (10, 35, 39) with A/(2*n) = 168/84 = 2, (14, 30, 40) with A/(2*n) = 168/84 = 2, (15, 34, 35) with A/(2*n) = 252/84 = 3 and (26, 28, 30) with A/(2*n) = 336/84 = 4 satisfy the condition.
a(426) = 0, because no triangle satisfies the condition. Therefore, a(n) = 0 for all n for which n*k = 426 for positive integers k.

Felix Huber, Table of n, a(n) for n = 1..500
Wikipedia, Heron's formula

Cf. A010814, A098030, A221837, A221838, A369951.

A370599 := proc(n) local u, v, w, A, q, i; i := 0; for u to floor(2/3*n) do for v from max(u, floor(n - u) + 1) to floor(n - 1/2*u) do w := 2*n - u - v; A := sqrt(n*(n - u)*(n - v)*(n - w)); if A = floor(A) then q := 1/2*A/n; if q = floor(q) then i := i + 1; end if; end if; end do; end do; return i; end proc;
seq(A370599(n), n = 1 .. 87);

a(n*k) >= a(n) for positive integers k.

A318575 Areas of primitive Heron triangles with square sides.

Original entry on oeis.org

32918611718880, 284239560530875680

Offset: 1

Max Alekseyev, Aug 29 2018

			a(1) is the area of the Heron triangle with sides 1853^2, 4380^2, 4427^2.
a(2) is the area of the Heron triangle with sides 11789^2, 68104^2, 68595^2.

P. Stanica, S. Sarkar, S. S. Gupta, S. Maitra, N. Kar. Counting Heron Triangles, Integers 13 (2013), A3.
R. Rathbun, Second primitive Heron triangle with square sides found, NMBRTHRY, 2018.

Cf. A046079, A221836, A221837, A221838, A223941, A237518.

a(1) was found by Stanica et al. (2013).

a(2) was found by Randall L Rathbun (2018).

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

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A370599 a(n) is the number of distinct triangles with integral side-lengths for which the perimeter 2*n divides the area.

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A318575 Areas of primitive Heron triangles with square sides.

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