A331256 -id:A331256 - cp's OEIS frontend

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.

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

Hugo Pfoertner, Jan 19 2020

nonn

			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

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

Hugo Pfoertner, Jan 19 2020

nonn

			See A331254.

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

Reinhard Zumkeller, May 05 2002

nonn

G. C. Greubel, Table of n, a(n) for the first 55 rows
Reinhard Zumkeller, Integer-sided triangles

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).

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 *)

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

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

Rick L. Shepherd, May 14 2008

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.

			2.06559111797728900542894154655061312577755824282218177418003934192719098236...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Circumradius.
Index entries for algebraic numbers, degree 2

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).

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

PARI
```
8/sqrt(15)
```

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

Hugo Pfoertner, Jan 20 2020

nonn

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.

			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

Cf. A128006, A128007, A192493, A192494, A331227, A331228, A331251, A331252, A331253, A331254, A331255, A331256.

Cf. A331245 (middle side), A331246 (longest side).

cp's OEIS Frontend

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.

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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.

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

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A140247 Decimal expansion of 8/sqrt(15).

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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.

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