A070093 Number of acute integer triangles with perimeter n.

0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 2, 4, 3, 5, 4, 5, 5, 5, 6, 6, 6, 7, 7, 9, 8, 10, 9, 10, 10, 11, 12, 12, 12, 14, 13, 16, 14, 17, 16, 17, 18, 18, 20, 20, 20, 22, 22, 24, 23, 25, 26, 26, 27, 28, 30, 30, 29, 32, 31, 35, 33, 36, 36, 38, 39, 40, 40

Offset: 1

Author

Reinhard Zumkeller, May 05 2002

Keywords

• nonn

Comments

An integer triangle [A070080(k) <= A070081(k) <= A070082(k)] is acute iff A070085(k) > 0.

Examples

			For n=9 there are A005044(9)=3 integer triangles: [1,4,4], [2,3,4] and [3,3,3]; two of them are acute, as 2^2+3^2<16=4^2, therefore a(9)=2.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Acute Triangle.
• R. Zumkeller, Integer-sided triangles

Crossrefs

Cf. A070094, A070095, A070118.

Cf. A005044, A024155, A070101.

Programs

Mathematica

Table[Sum[Sum[(1 - Sign[Floor[(n - i - k)^2/(i^2 + k^2)]]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}] (* Wesley Ivan Hurt, May 12 2019 *)

Formula

a(n) = A005044(n) - A070101(n) - A024155(n);

a(n) = A042154(n) + A070098(n).

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1-sign(floor((n-i-k)^2/(i^2+k^2)))) * sign(floor((i+k)/(n-i-k+1))). - Wesley Ivan Hurt, May 12 2019