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%I #26 Mar 30 2024 12:26:02
%S 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,
%T 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,
%U 0,3,0,2,0,0,2,2,3,1,0,2,4,1,0,8,1,0,1
%N a(n) is the number of distinct triangles with integral side-lengths for which the perimeter 2*n divides the area.
%C 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.
%C 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.
%H Felix Huber, <a href="/A370599/b370599.txt">Table of n, a(n) for n = 1..500</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Heron%27s_formula">Heron's formula</a>
%F a(n*k) >= a(n) for positive integers k.
%e 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).
%e 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.
%e 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.
%p 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;
%p seq(A370599(n), n = 1 .. 87);
%Y Cf. A010814, A098030, A221837, A221838, A369951.
%K nonn,easy
%O 1,21
%A _Felix Huber_, Mar 09 2024
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