A070106 -id:A070106 - cp's OEIS frontend

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

Reinhard Zumkeller, May 05 2002

nonn

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

			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.

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

Cf. A070102, A070103, A070127.

Cf. A005044, A024155, A070093.

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 *)

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

A070098 Number of integer triangles with perimeter n which are acute and isosceles.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 9, 10, 10, 11, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 14, 15, 15, 16, 15

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

Equivalently, the number of obtuse isosceles integer triangles with base n. - Charlie Marion, Jun 18 2019

			For n=9 there are A005044(9)=3 integer triangles: [1,4,4], [2,3,4] and [3,3,3]; both isosceles are also acute.

Marius A. Burtea, Table of n, a(n) for n = 1..10000
Reinhard Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A059169, A070099, A070100, A070124.

[Floor(k/2)-Floor(k/(2 + Sqrt(2)))-((k + 1) mod 2): k in [1..76]]; // Marius A. Burtea, Jun 21 2019

a(n) = A070093(n)-A024154(n); a(n) = A059169(n)-A070106(n).
a(n) = floor(n/2) - floor(n/(2 + sqrt(2))) - ((n + 1) mod 2). - David Pasino, Jun 27 2016
a(n) = A004526(n-1) - A183138(n). - R. J. Mathar, May 22 2019

A070108 Number of integer triangles with perimeter n and prime side lengths which are obtuse and isosceles.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(k)<=1 until k = 140, for k = 141 there are A005044(141)=432 integer triangles, a(141)=2 as
[37=37<67]: 37+37+67 = 141 and 2*(37^2)<67^2 and 37, 67 are primes,
[41=41<59]: 41+41+59 = 141 and 2*(41^2)<59^2 and 41, 59 are primes.

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070101, A059169, A070088, A070092, A070103, A070106, A070135.

A070107 Number of integer triangles with perimeter n and relatively prime side lengths which are obtuse and isosceles.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 2, 1, 0, 1, 1, 0, 1, 1, 2, 1, 2, 1, 2, 0, 1, 1, 2, 1, 1, 1, 2, 1, 2, 0, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 1, 3, 1, 3, 2, 1, 1, 1, 0, 4, 2, 2, 2, 4, 1, 3, 2, 3, 2, 4, 1

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070101, A059169, A051493, A070091, A070102, A070106, A070084, A070134.

A070133 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse isosceles integer triangle.

Original entry on oeis.org

5, 14, 26, 32, 52, 61, 82, 91, 104, 118, 133, 146, 163, 182, 202, 219, 242, 246, 266, 291, 314, 342, 347, 372, 404, 432, 437, 467, 472, 504, 542, 547, 577, 582, 619, 625, 663, 709, 714, 751, 757, 801, 807, 853, 858, 907, 913

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(5)=52: [A070080(52), A070081(52), A070082(52)]=[5=5<8], A070085(52)=5^2+5^2-8^2=25+25-64=-14<0.

R. Zumkeller, Integer-sided triangles

Cf. A070106, A070134, A070135, A070115, A070127.

cp's OEIS Frontend

A070101 Number of obtuse integer triangles with perimeter n.

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A070098 Number of integer triangles with perimeter n which are acute and isosceles.

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A070108 Number of integer triangles with perimeter n and prime side lengths which are obtuse and isosceles.

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A070107 Number of integer triangles with perimeter n and relatively prime side lengths which are obtuse and isosceles.

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A070133 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse isosceles integer triangle.

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