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

1,9

Comments

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

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

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

Example

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.

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 *)

Crossrefs

Cf. A070094, A070095, A070118.

Cf. A005044, A024155, A070101.

Sequence in context: A293223 A361514 A029348 * A174931 A058744 A366297

Adjacent sequences: A070090 A070091 A070092 * A070094 A070095 A070096

Keyword

nonn

Author

Reinhard Zumkeller, May 05 2002

Status

approved