A070101 Number of obtuse integer triangles with perimeter n.

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 2, 3, 2, 3, 3, 5, 3, 7, 4, 8, 5, 9, 7, 10, 8, 11, 9, 14, 11, 16, 12, 18, 14, 19, 17, 21, 18, 23, 21, 27, 22, 30, 24, 32, 27, 34, 30, 37, 33, 40, 35, 44, 37, 47, 40, 50, 44, 53, 49, 56, 52, 60, 55, 64, 57, 68

Offset

1,11

Comments

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

Links

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Obtuse Triangle.

R. Zumkeller, Integer-sided triangles

Formula

a(n) = A005044(n) - A070093(n) - A024155(n).

a(n) = A024156(n) + A070106(n).

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)}

(1-sign(floor((i^2 + k^2)/(n-i-k)^2))) * sign(floor((i+k)/(n-i-k+1))). - Wesley Ivan Hurt, May 12 2019

Example

For n=14 there are A005044(14)=4 integer triangles: [2,6,6], [3,5,6], [4,4,6] and [4,5,5]; two of them are obtuse, as 3^2+5^2<36=6^2 and 4^2+4^2<36=6^2, therefore a(14)=2.

Mathematica

Table[Sum[Sum[(1 - Sign[Floor[(i^2 + k^2)/(n - i - 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. A070102, A070103, A070127.

Cf. A005044, A024155, A070093.

Sequence in context: A288311 A244366 A262676 * A022830 A035663 A117192

Adjacent sequences: A070098 A070099 A070100 * A070102 A070103 A070104

Keyword

nonn

Author

Reinhard Zumkeller, May 05 2002

Status

approved