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 1-10 of 15 results. Next

A331227 a(n) = numerator of squared radius R^2 of the circumcircle of the n-th triangle with integer sides i <= j <= k in a list of such triangles, the list being sorted by increasing size of R. Denominators are A331228.

Original entry on oeis.org

1, 16, 4, 16, 81, 81, 3, 81, 256, 64, 64, 256, 16, 25, 625, 625, 256, 625, 1600, 81, 625, 25, 2025, 1296, 64, 64, 324, 48, 625, 81, 400, 1296, 5184, 12, 625, 3136, 2401, 3969, 2401, 1225, 1225, 2401, 2401, 1296, 2401, 784, 2401, 50176, 6400, 4096, 81, 1024, 49, 49, 49, 49
Offset: 1

Views

Author

Hugo Pfoertner, Jan 12 2020

Keywords

Comments

Radii shared by more than one triangle are not removed. The first occurrence is for squared radius 64/15 at positions n = 10 and n = 11.

Examples

			The first terms b(n) = a(n)/A331228(n) correspond to the following triangles (i, j, k):
  b(1) = 1/3: (1,1,1),
  b(2) = 16/15: (1,2,2),
  b(3) = 4/3: (2,2,2),
  b(4) = 16/7: (2,2,3) (obtuse triangle excluded in A331222),
  b(5) = 81/35: (1,3,3),
  b(6) = 81/32: (2,3,3),
  b(7) = 3/1: (3,3,3),
  b(8) = 81/20: (3,3,4),
  b(9) = 256/63: (1,4,4),
  b(10) = 64/15: (2,3,4), (obtuse)
  b(11) = 64/15: (2,4,4).

Crossrefs

Cf. A331222, A331223, A331228.

Formula

Squared radius of circumcircle of triangle with sides a, b, c:
R^2 = (a*b*c)^2 / (16*s*(s - a)*(s - b)*(s - c)) with s = (a + b + c)/2.

A140247 Decimal expansion of 8/sqrt(15).

Original entry on oeis.org

2, 0, 6, 5, 5, 9, 1, 1, 1, 7, 9, 7, 7, 2, 8, 9, 0, 0, 5, 4, 2, 8, 9, 4, 1, 5, 4, 6, 5, 5, 0, 6, 1, 3, 1, 2, 5, 7, 7, 7, 5, 5, 8, 2, 4, 2, 8, 2, 2, 1, 8, 1, 7, 7, 4, 1, 8, 0, 0, 3, 9, 3, 4, 1, 9, 2, 7, 1, 9, 0, 9, 8, 2, 3, 6, 6, 3, 8, 8, 8, 1, 7, 8, 7, 6, 9, 5, 3, 2, 6, 7, 6, 5, 7, 9, 5, 9, 2, 0, 9, 5, 5, 5, 3, 6
Offset: 1

Views

Author

Rick L. Shepherd, May 14 2008

Keywords

Comments

Circumradius of the obtuse scalene triangle with sides of lengths 2, 3 and 4, the scalene triangle with least integer side lengths. See formulas in the Weisstein link.

Examples

			2.06559111797728900542894154655061312577755824282218177418003934192719098236...

Links

Crossrefs

Cf. A140239, A140240, A140241, A140242, A140243, A140244, A140245, A140246, A140248, A140249, A088543, A010472.
Equals sqrt(A331227(10)/A331228(10)) = sqrt(A331227(11)/A331228(11)), A331254, A331255, A331256 (list of triangles with integer sides sorted by circumradius).

Programs

  • Mathematica
    RealDigits[8/Sqrt[15],10,120][[1]] (* Harvey P. Dale, May 06 2012 *)
  • PARI
    8/sqrt(15)

Formula

8/sqrt(15) = 8/A010472.

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).

A331254 Triangles with integer sides i <= j <= k sorted by radius of circumcircle, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives the shortest side i. The other sides are in A331255 and A331256.

Original entry on oeis.org

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

Views

Author

Hugo Pfoertner, Jan 19 2020

Keywords

Examples

			List of triangles begins:
   n
   |     R^2 = A331227(n)/A331228(n)
   |     |    i .... (this sequence)
   |     |    | j .. (A331255)
   |     |    | | k  (A331256)
   |     |    | | |
   1    1/ 3  1 1 1
   2   16/15  1 2 2
   3    4/ 3  2 2 2
   4   16/ 7  2 2 3
   5   81/35  1 3 3
   6   81/32  2 3 3
   7    3/ 1  3 3 3
   8   81/20  3 3 4
   9  256/63  1 4 4
  10   64/15  2 3 4
  11   64/15  2 4 4
  12  256/55  3 4 4

Crossrefs

Cf. A331227, A331228, A331251, A331252, A331253.
Cf. A331255 (middle side), A331256 (longest side).

A331255 Triangles with integer sides i <= j <= k sorted by radius of circumcircle, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives the middle side j. The other sides are in A331254 and A331256.

Original entry on oeis.org

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

Views

Author

Hugo Pfoertner, Jan 19 2020

Keywords

Examples

			See A331254.

Crossrefs

Cf. A331227, A331228 (squared radius of circumcircle), A331254 (shortest side), A331256 (longest side).

A331256 Triangles with integer sides i <= j <= k sorted by radius of circumcircle, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives the longest side k. The other sides are in A331254 and A331255.

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, 8, 8, 8, 8, 8, 7, 8, 8, 7, 8, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 8, 9, 9, 9
Offset: 1

Views

Author

Hugo Pfoertner, Jan 19 2020

Keywords

Examples

			See A331254.

Crossrefs

Cf. A331227, A331228 (squared radius of circumcircle), A331254 (shortest side), A331255 (middle side).

A331222 a(n) = numerator of squared radius R^2 of the circumcircle of the n-th non-obtuse triangle with integer sides i <= j <= k <= sqrt(i^2 + j^2) in a list of such triangles, the list being sorted by increasing size of R. Denominators are A331223.

Original entry on oeis.org

1, 16, 4, 81, 81, 3, 81, 256, 64, 256, 16, 25, 625, 625, 256, 625, 625, 25, 1296, 64, 324, 48, 625, 81, 1296, 12, 625, 3136, 2401, 2401, 1225, 2401, 2401, 1296, 2401, 2401, 50176, 4096, 81, 1024, 49, 49, 4096, 256, 256, 4096, 2401, 1024, 4096, 35721, 6561
Offset: 1

Views

Author

Hugo Pfoertner, Jan 12 2020

Keywords

Comments

Radii shared by more than one triangle are not removed. The first occurrence is for squared radius 49/3 at positions n = 41 and n = 42.

Examples

			The first terms b(n) = a(n)/A331223(n) correspond to the following triangles (i, j, k):
  b(1) = 1/3: (1,1,1),
  b(2) = 16/15: (1,2,2),
  b(3) = 4/3: (2,2,2),
  b(4) = 81/35: (1,3,3),
  b(5) = 81/32: (2,3,3),
  b(6) = 3/1: (3,3,3),
  b(7) = 81/20: (3,3,4),
  b(8) = 256/63: (1,4,4),
  b(9) = 64/15: (2,4,4),
...
  b(41) = b(42) = 49/3: (5,7,8), (7,7,7).

Crossrefs

Cf. A317182, A331223, A331227, A331228.

Formula

Squared radius of circumcircle of triangle with sides a, b, c:
R^2 = (a*b*c)^2 / (16*s*(s - a)*(s - b)*(s - c)) with s = (a + b + c)/2.

A331223 a(n) = denominator of squared radius R^2 of the circumcircle of the n-th non-obtuse triangle with integer sides i <= j <= k <= sqrt(i^2 + j^2) in a list of such triangles, the list being sorted by increasing size of R. Numerators are A331222.

Original entry on oeis.org

3, 15, 3, 35, 32, 1, 20, 63, 15, 55, 3, 4, 99, 96, 39, 91, 84, 3, 143, 7, 35, 5, 64, 8, 119, 1, 51, 255, 195, 192, 96, 187, 180, 95, 171, 160, 3135, 255, 5, 63, 3, 3, 247, 15, 15, 231, 132, 55, 207, 1760, 323, 11, 320, 35, 115, 8855, 308, 3, 299, 20
Offset: 1

Views

Author

Hugo Pfoertner, Jan 12 2020

Keywords

Comments

See A331222.

Examples

			See A331222.

Crossrefs

Cf. A331222, A331227, A331228.

A331229 a(n) = number of triangles with integer sides i <= j <= k with radius of circumcircle <= n.

Original entry on oeis.org

1, 7, 22, 47, 91, 148, 231, 334, 469, 631, 830, 1062, 1339, 1657, 2024, 2434, 2905, 3427, 4014, 4653, 5362, 6141, 6994, 7911, 8917, 10000, 11169, 12425, 13774, 15211, 16743, 18381, 20133, 21975, 23929, 25998, 28185, 30482, 32906, 35449, 38137, 40935, 43884, 46954
Offset: 1

Views

Author

Hugo Pfoertner, Jan 13 2020

Keywords

Examples

			The radius of the m-th circumcircle in the sorted list is R(m) = sqrt(A331227(m)/A331228(m)). The list of radii, rounded to 10^-4, starts: {0.57735, 1.0328, 1.1547, 1.5119, 1.5213, 1.5910, 1.7321, 2.0125, 2.0158, 2.0656, 2.0656, 2.1574, 2.3094, 2.5000, 2.5126, 2.5516, 2.5621, 2.6207, 2.6318, 2.7136, 2.7277, 2.8868, 3.0067, ...}.
a(1) = 1: 1 circle (R = 0.57735) with R <= 1,
a(2) = 7: a(1) + 6 circles (R = 1.0328, 1.1547, 1.5119, 1.5213, 1.5910, 1.7321) with 1 < R <= 2,
a(3) = 22: a(2) + 15 circles (R = 2.0125, 2.0158, 2.0656, 2.0656, 2.1574, 2.3094, 2.5000, 2.5126, 2.5516, 2.5621, 2.6207, 2.6318, 2.7136, 2.7277, 2.8868) with 2 < R <= 3.

Crossrefs

Cf. A331227, A331228.
Bisection of A331240 (n even).

A331240 a(n) = number of triangles with integer sides i <= j <= k with diameter of circumcircle <= n.

Original entry on oeis.org

0, 1, 3, 7, 14, 22, 34, 47, 67, 91, 117, 148, 187, 231, 281, 334, 400, 469, 548, 631, 727, 830, 943, 1062, 1202, 1339, 1490, 1657, 1833, 2024, 2226, 2434, 2662, 2905, 3155, 3427, 3712, 4014, 4321, 4653, 5005, 5362, 5749, 6141, 6558, 6994, 7440, 7911, 8408, 8917
Offset: 1

Views

Author

Hugo Pfoertner, Jan 13 2020

Keywords

Examples

			The diameter of the n-th circumcircle in the sorted list is D(n) = 2*sqrt(A331227(n)/A331228(n)). The list of diameters, rounded to 10^-4, starts: {1.1547, 2.0656, 2.3094, 3.0237, 3.0426, 3.1820, 3.4641, 4.0249, 4.0316, 4.1312, 4.1312, 4.3149, 4.6188, 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) = 3: a(2) + 2 circles (D = 2.0656, 2.3094) with 2 < D <= 3,
a(4) = 7: a(3) + 4 circles (D = 3.02, 3.04, 3.18, 3.46) with 3 < D <= 4,
a(5) = 14: a(4) + 7 circles (D = 4.0249, ..., 5) with 4 < D <= 5.

Crossrefs

Cf. A331227, A331228, A331229.
Showing 1-10 of 15 results. Next

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