A024154 -id:A024154 - cp's OEIS frontend

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

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

A024156 Number of integer-sided triangles with sides a,b,c, a

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 1, 2, 2, 3, 2, 4, 3, 6, 3, 7, 4, 8, 6, 9, 7, 10, 8, 12, 10, 15, 11, 16, 13, 18, 15, 19, 17, 21, 19, 25, 21, 28, 22, 30, 25, 32, 28, 35, 31, 38, 33, 41, 35, 45, 38, 47, 42, 51, 46, 53, 50, 57, 52, 61, 55, 65, 58, 68, 63, 74, 67, 77, 71, 82, 74, 86, 77, 92, 82, 96, 87

Offset: 1

Clark Kimberling

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..200

Cf. A024154, A024155.

r[n_] := Module[{a, b}, Reduce[a>1 && a n && 2n(a + b) < 2a*b + n^2, {a, b}, Integers]];
a[n_] := Module[{rn = r[n]}, Which[rn === False, 0, rn[[0]] === And, 1, rn[[0]] === Or, Length[rn], True, Print["error: ", n, " ", rn]]];
Array[a, 84] (* Jean-François Alcover, Mar 01 2019 *)

A070097 Number of integer triangles with perimeter n and prime side lengths which are both acute and scalene.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070093, A005044, A070088, A070090, A070095, A024154, A070123.

A070096 Number of integer triangles with perimeter n and relatively prime side lengths which are both acute and scalene.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 3, 4, 2, 4, 4, 4, 4, 5, 4, 6, 5, 8, 5, 9, 7, 8, 7, 8, 10, 10, 9, 12, 10, 12, 11, 14, 11, 15, 13, 15, 14, 18, 13, 19, 15, 21, 16, 23, 17, 24, 20, 25, 21, 28, 20, 28, 23, 28

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070093, A005044, A051493, A070094, A024154, A070084, A070122.

A070121 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute scalene integer triangle.

Original entry on oeis.org

33, 45, 53, 60, 70, 83, 90, 92, 106, 114, 119, 132, 134, 142, 148, 162, 165, 168, 175, 181, 183, 197, 200, 203, 204, 218, 221, 224, 237, 240, 245, 247, 261, 264, 267, 268, 282, 290, 293, 296, 309, 312, 316, 317, 319, 333, 341

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(4)=60: [A070080(60), A070081(60), A070082(60)]=[4<7<8], A070085(60)=4^2+7^2-8^2=16+49-64=1>0.

R. Zumkeller, Integer-sided triangles

Cf. A024154, A070122, A070123, A070112, A070118.

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

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A024156 Number of integer-sided triangles with sides a,b,c, a

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A070097 Number of integer triangles with perimeter n and prime side lengths which are both acute and scalene.

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A070096 Number of integer triangles with perimeter n and relatively prime side lengths which are both acute and scalene.

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A070121 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute scalene integer triangle.

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