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

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

Author

Felix Huber, Mar 09 2024

Keywords

• nonn
• easy

Comments

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.

Examples

			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.
		

Links

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

Crossrefs

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

Programs

Maple

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);

Formula

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