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 1-3 of 3 results.

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.

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.

A331228 a(n) = denominator 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. Numerators are A331227.

Original entry on oeis.org

3, 15, 3, 7, 35, 32, 1, 20, 63, 15, 15, 55, 3, 4, 99, 96, 39, 91, 231, 11, 84, 3, 224, 143, 7, 7, 35, 5, 64, 8, 39, 119, 455, 1, 51, 255, 195, 320, 192, 96, 96, 187, 180, 95, 171, 55, 160, 3135, 399, 255, 5, 63, 3, 3, 3, 3, 247, 15, 15, 15, 15, 36, 231, 132, 55
Offset: 1

Views

Author

Hugo Pfoertner, Jan 12 2020

Keywords

Comments

See A331222 and A331227.

Examples

			See A331227.

Crossrefs

Cf. A331222, A331223, A331227.
Showing 1-3 of 3 results.

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