A070124 -id:A070124 - cp's OEIS frontend

A070118 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute integer triangle.

1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 15, 16, 18, 19, 22, 23, 24, 27, 28, 31, 33, 34, 35, 38, 39, 40, 43, 45, 46, 47, 48, 51, 53, 54, 55, 58, 60, 63, 64, 65, 68, 70, 71, 72, 73, 76, 81, 83, 84, 85, 88, 90, 92, 93, 94, 95, 98, 103, 106, 107, 108

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(21)=33: [A070080(33), A070081(33), A070082(33)]=[4,5,6], A070085(33)=4^2+5^2-6^2=16+25-36=5>0.

Jean-François Alcover, Table of n, a(n) for n = 1..431
Reinhard Zumkeller, Integer-sided triangles

Cf. A070093, A070119, A070120, A070121, A070122, A070123, A070124, A070125, A070126.

m = 50; (* max perimeter *)
sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];
triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1] & // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]] &];
Position[triangles, {a_, b_, c_} /; a^2 + b^2 - c^2 > 0] // Flatten (* Jean-François Alcover, Oct 04 2021 *)

A070115 Numbers m such that [A070080(m), A070081(m), A070082(m)] is an isosceles integer triangle.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 15, 16, 18, 19, 22, 23, 24, 26, 27, 28, 31, 32, 34, 35, 38, 39, 40, 43, 46, 47, 48, 51, 52, 54, 55, 58, 61, 63, 64, 65, 68, 71, 72, 73, 76, 81, 82, 84, 85, 88, 91, 93, 94, 95, 98, 103, 104, 107, 108

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			26 is a term because [A070080(26), A070081(26), A070082(26)] = [4=4<6].

Jean-François Alcover, Table of n, a(n) for n = 1..365
Reinhard Zumkeller, Integer-sided triangles

Cf. A059169, A070116, A070117, A070124, A070125, A070126, A070133, A070134, A070135.

m = 55 (* max perimeter *);
sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];
triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1]& // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]] &];
Position[triangles, {a_, a_, b_} | {a_, b_, b_}] // Flatten (* Jean-François Alcover, Oct 12 2021 *)

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

cp's OEIS Frontend

A070118 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute integer triangle.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A070115 Numbers m such that [A070080(m), A070081(m), A070082(m)] is an isosceles integer triangle.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Formula