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Number of acute integer triangles with perimeter n.
21

%I #13 May 13 2019 04:25:35

%S 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,

%T 11,12,12,12,14,13,16,14,17,16,17,18,18,20,20,20,22,22,24,23,25,26,26,

%U 27,28,30,30,29,32,31,35,33,36,36,38,39,40,40

%N Number of acute integer triangles with perimeter n.

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

%H Seiichi Manyama, <a href="/A070093/b070093.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AcuteTriangle.html">Acute Triangle</a>.

%H R. Zumkeller, <a href="/A070080/a070080.txt">Integer-sided triangles</a>

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

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

%F 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

%e 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.

%t 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 *)

%Y Cf. A070094, A070095, A070118.

%Y Cf. A005044, A024155, A070101.

%K nonn

%O 1,9

%A _Reinhard Zumkeller_, May 05 2002