cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A331244 Triangles with integer sides i <= j <= k sorted by radius of enclosing circle, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives the shortest side i. The other sides are in A331245 and A331246.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 3, 2, 3, 1, 2, 3, 4, 2, 3, 3, 1, 2, 4, 3, 4, 5, 2, 3, 3, 4, 1, 4, 2, 3, 5, 4, 5, 6, 2, 3, 3, 4, 4, 5, 4, 1, 2, 5, 3, 4, 6, 5, 6, 2, 3, 3, 4, 4, 5, 5, 4, 1, 6, 2, 5, 7, 3, 4, 6, 5, 7, 6, 7, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 1, 5, 2, 3, 7, 6, 4
Offset: 1

Views

Author

Hugo Pfoertner, Jan 20 2020

Keywords

Comments

The enclosing circle differs from the circumcircle by limiting the radius to (longest side)/2 for obtuse triangles, i.e., those with i^2 + j^2 < k^2.

Examples

			List of triangles begins:
   n
   |     R^2
   |     |    i .... (this sequence)
   |     |    | j .. (A331245)
   |     |    | | k  (A331246)
   |     |    | | |
   1    1/ 3  1 1 1
   2   16/15  1 2 2
   3    4/ 3  2 2 2
   4    9/ 4  2 2 3  obtuse
   5   81/35  1 3 3
   6   81/32  2 3 3
   7    3/ 1  3 3 3
   8    4/ 1  2 3 4  obtuse
   9   81/20  3 3 4
  10  256/63  1 4 4
  11   64/15  2 4 4
  12  256/55  3 4 4
  13   16/ 3  4 4 4
  14   25/ 4  2 4 5  obtuse
  15   25/ 4  3 3 5  obtuse
  16   25/ 4  3 4 5
  17  625/99  1 5 5

Crossrefs

Cf. A128006, A128007, A192493, A192494, A331227, A331228, A331251, A331252, A331253, A331254, A331255, A331256.
Cf. A331245 (middle side), A331246 (longest side).

A331246 Triangles with integer sides i <= j <= k sorted by radius of enclosing circle, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives the longest side k. The other sides are in A331244 and A331245.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9
Offset: 1

Views

Author

Hugo Pfoertner, Jan 20 2020

Keywords

Comments

The enclosing circle differs from the circumcircle by limiting the radius to (longest side)/2 for obtuse triangles, i.e., those with i^2 + j^2 < k^2.

Examples

			See A331244.

Crossrefs

Cf. A331244 (shortest side), A331245 (middle side).

A070080 Smallest side of integer triangles [a(n) <= A070081(n) <= A070082(n)], sorted by perimeter, lexicographically ordered.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 2, 3, 2, 3, 1, 2, 3, 3, 2, 3, 4, 1, 2, 3, 3, 4, 2, 3, 4, 4, 1, 2, 3, 3, 4, 4, 5, 2, 3, 4, 4, 5, 1, 2, 3, 3, 4, 4, 5, 5, 2, 3, 4, 4, 5, 5, 6, 1, 2, 3, 3, 4, 4, 5, 5, 5, 6, 2, 3, 4, 4, 5, 5, 6, 6, 1, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 2, 3, 4, 4, 5, 5
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Links

Crossrefs

Cf. A046128, A055594, A069597.
Cf. A070081, A070082, A070083, A070084, A070085, A070086.
Cf. A316841, A316843, A316844, A316845 (sides (i,j,k) with j + k > i >= j >= k >= 1).
Cf. A331244, A331245, A331246 (similar, but triangles sorted by radius of enclosing circle), A331251, A331252, A331253 (triangles sorted by area), A331254, A331255, A331256 (triangles sorted by radius of circumcircle).

Programs

  • Mathematica
    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]]&];
triangles[[All, 1]] (* Jean-François Alcover, Jun 12 2012, updated Jul 09 2017 *)

Formula

a(n) = A070083(n) - A070082(n) - A070081(n).

A331242 a(n) = number of triangles with integer sides i <= j <= k with radius of enclosing circle <= n.

Original entry on oeis.org

1, 8, 26, 56, 106, 175, 272, 397, 555, 750
Offset: 1

Views

Author

Hugo Pfoertner, Jan 20 2020

Keywords

Comments

The enclosing circle differs from the circumcircle by limiting the radius to (longest side)/2 for obtuse triangles, i.e., those with i^2 + j^2 < k^2.

Examples

			The list of radii of the n-th enclosing circle, rounded to 10^-4, starts: {0.57735, 1.0328, 1.1547, 1.5000, 1.5213, 1.5910, 1.7321, 2.0000, 2.0125, 2.0158, 2.0656, 2.1574, 2.3094, 2.5000, 2.5000, 2.5000, 2.5126, 2.5516, 2.5621, 2.6207, 2.7277, 2.8868, 3.0000, 3.0000, 3.0000, 3.0000, 3.0105, ...}.
a(1) = 1: 1 circle (R = 0.57735) with R <= 1,
a(2) = 8: a(1) + 7 circles (R = 1.0328, 1.1547, 1.5000, 1.5213, 1.5910, 1.7321, 2.0000) with 1 < R <= 2,
a(3) = 26: a(2) + 18 circles (R = 2.0125, 2.0158, 2.0656, 2.1574, 2.3094, 2.5000, 2.5000, 2.5000, 2.5126, 2.5516, 2.5621, 2.6207, 2.7277, 2.8868, 3.0000, 3.0000, 3.0000, 3.0000) with 2 < R <= 3.

Crossrefs

Cf. A331229, A331240, A331243, A331244, A331245, A331246.

A331243 a(n) = number of triangles with integer sides i <= j <= k with diameter of enclosing circle <= n.

Original entry on oeis.org

0, 1, 4, 8, 16, 26, 39, 56, 79, 106, 138, 175, 221, 272, 331, 397, 471, 555, 648, 750
Offset: 1

Views

Author

Hugo Pfoertner, Jan 20 2020

Keywords

Comments

The enclosing circle differs from the circumcircle by limiting the diameter to the longest side k for obtuse triangles, i.e., those with i^2 + j^2 < k^2.

Examples

			The sorted list of diameters D(n), rounded to 10^-4, starts: {1.1547, 2.0656, 2.3094, 3.0000, 3.0426, 3.1820, 3.4641, 4.0000, 4.0249, 4.0316, 4.1312, 4.3149, 4.6188, 5.0000, 5.0000, 5.0000, 5.0252, ...}.
a(1) = 0: 0 circles with D <= 1,
a(2) = 1: 1 circle (D = 1.1547) with 1 < D <= 2,
a(3) = 4: a(2) + 3 circles (D = 2.0656, 2.3094, 3.0000) with 2 < D <= 3,
a(4) = 8: a(3) + 4 circles (D = 3.04, 3.18, 3.46, 4.00) with 3 < D <= 4,
a(5) = 16: a(4) + 8 circles (D = 4.0249, ..., 5, 5, 5) with 4 < D <= 5.

Crossrefs

Cf. A331229, A331240, A331242, A331244, A331245, A331246.
Showing 1-5 of 5 results.

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