A331253 -id:A331253 - cp's OEIS frontend

A331251 Triangles with integer sides i <= j <= k sorted by area, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives shortest side i. The other sides are in A331252 and A331253.

1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 2, 3, 1, 3, 3, 1, 2, 2, 1, 3, 1, 2, 3, 2, 1, 3, 2, 1, 3, 4, 2, 4, 1, 3, 2, 3, 1, 3, 4, 2, 4, 1, 2, 4, 1, 3, 3, 2, 3, 1, 2, 4, 1, 4, 4, 5, 2, 3, 4, 2, 1, 3, 3, 1, 2, 5, 4, 2, 1, 3, 4, 5, 1, 4, 2, 3, 3, 2, 4, 1, 5, 5, 3, 4, 1, 5

Offset: 1

Hugo Pfoertner, Jan 19 2020

nonn

			List of first triangles:
   n
   | 16*A^2
   |    | i .... (this sequence)
   |    | | j .. (A331252)
   |    | | | k  (A331253)
   |    | | | |
   1    3 1 1 1
   2   15 1 2 2
   3   35 1 3 3
   4   48 2 2 2
   5   63 1 4 4
   6   63 2 2 3
   7   99 1 5 5
   8  128 2 3 3
   9  135 2 3 4
  10  143 1 6 6
  11  195 1 7 7

Cf. A316841, A316843, A316844, A316845, A316853, A317182, A331011.

Cf. A331252 (middle side j), A331253 (longest side k).

A331252 Triangles with integer sides i <= j <= k sorted by area, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives middle side j. The other sides are in A331251 and A331253.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 5, 3, 3, 6, 7, 4, 4, 3, 8, 3, 3, 9, 5, 5, 10, 4, 11, 6, 4, 6, 12, 4, 7, 13, 5, 4, 7, 4, 14, 5, 8, 5, 15, 6, 4, 8, 4, 16, 9, 5, 17, 6, 7, 9, 6, 18, 10, 5, 19, 6, 5, 5, 11, 8, 5, 10, 20, 7, 7, 21, 12, 5, 7, 11, 22, 9, 6, 6, 23, 6, 13, 8, 8, 12

Offset: 1

Hugo Pfoertner, Jan 19 2020

nonn

			See A331251.

Cf. A331251 (shortest side), A331253 (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).

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).

A331250 a(n) = number of triangles with integer sides i <= j <= k with area <= n.

Original entry on oeis.org

2, 6, 10, 15, 21, 28, 35, 44, 52, 63, 71, 84, 92, 105, 118, 128, 143, 159, 173, 183, 200, 214, 231, 248, 264, 280, 301, 316, 332, 356, 370, 394, 414, 428, 451, 475, 494, 514, 535, 557, 580, 607, 624, 645, 678, 697, 718, 748, 770, 794, 822, 845, 873, 900, 927

Offset: 1

Hugo Pfoertner, Jan 20 2020

nonn

			The sorted list of areas A_k = A(A331251(k), A331252(k), A331253(k)), rounded to 10^-4, starts:: {0.43301, 0.96825, 1.4790, 1.7321, 1.9843, 1.9843, 2.4875, 2.8284, 2.9047, 2.9896, 3.4911, 3.7997, 3.8730, 3.8971, 3.9922, 4.1458, 4.4721, 4.4931, 4.6837, 4.8990, 4.9937, 5.3327, ...}.
a(1) = 2: 2 triangles (A = 0.43301, 0.96825) with A <= 1,
a(2) = 6: a(1) + 4 triangles (A = 1.4790, 1.7321, 1.9843, 1.9843) with 1 < A <= 2,
a(3) = 10: a(2) + 4 triangles (A = 2.4875, 2.8284, 2.9047, 2.9896) with 2 < A <= 3,
a(4) = 15: a(3) + 5 triangles (A = 3.4911, 3.7997, 3.8730, 3.8971, 3.9922) with 3 < A <= 4,
a(5) = 21: a(4) + 6 triangles (A = 4.1458, 4.4721, 4.4931, 4.6837, 4.8990, 4.9937) with 4 < A <= 5.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A316853, A317182, A331011, A331251, A331252, A331253.

from itertools import count
def A331250(n):
    m, c = n**2<<4, 0
    for k in count(1):
        if (k**2<<2) - 1 > m:
            break
        for j in range((k>>1)+1,k+1):
            for i in range(k-j+1,j+1):
                if ((-i + j + k)*(i - j + k)*(i + j - k)*(i + j + k)) > m:
                    break
                c += 1
    return c # Chai Wah Wu, Aug 25 2023

Area A of a triangle with sides a, b, c:

A(a, b, c) = sqrt(s*(s - a)*(s - b)*(s - c)) with s = (a + b + c)/2.

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

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).

A331247 Numerator of the x-coordinate of the 3rd point (x,y) of the n-th triangle with integer sides i <= j <= k in a list sorted by increasing area, when the triangle is drawn with the shortest side from (0,0) to (0,i) and the middle side from (0,i) to (x,y). x = a(n)/A331248(n), y = sqrt(A331249(n))/A331248(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 9, 1, 1, 11, 1, 1, 13, 1, 3, 1, 25, 8, 1, 15, 1, 1, 29, 1, 17, 3, 1, 1, 3, 19, 1, 11, 49, 1, 2, 1, 3, 21, 10, 1, 37, 25, 1, 9, 1, 23, 55, 1, 3, 41, 1, 11, 1, 25, 2, 1, 61, 5, 81, 27, 15, 27, 1, 1, 3, 4, 1, 29, 5, 67, 1, 1, 49, 2, 89, 1, 11, 31, 3

Offset: 1

Hugo Pfoertner, Jan 23 2020

The side lengths (i,j,k) of the triangles in the list sorted by area are given in A331251, A331252, and A331253.

			x(n) = a(n) / A331248(n),
y(n) = sqrt(A331249(n)) / A331248(n),
   n i (A331251)
   | | j (A331252)
   | | | k (A331253)
   | | | |    A^2*16
   | | | |    |  a(n) this sequence
   | | | |    |  |  A331248
   | | | |    |  |  |   A331249
   | | | |    |  |  |   |  (x, y)
   1 1 1 1    3  1  2   3  (0.5000, 0.86603)
   2 1 2 2   15  1  2  15  (0.5000, 1.9365)
   3 1 3 3   35  1  2  35  (0.5000, 2.9580)
   4 2 2 2   48  1  1   3  (1.0000, 1.7321)
   5 1 4 4   63  1  2  63  (0.5000, 3.9686)
   6 2 2 3   63  9  4  63  (2.2500, 1.9843)
   7 1 5 5   99  1  2  99  (0.5000, 4.9749)
   8 2 3 3  128  1  1   8  (1.0000, 2.8284)
   9 2 3 4  135 11  4 135  (2.7500, 2.9047)
  10 1 6 6  143  1  2 143  (0.5000, 5.9791)
  11 1 7 7  195  1  2 195  (0.5000, 6.9821)
  12 2 4 5  231 13  4 231  (3.2500, 3.7997)
  13 2 4 4  240  1  1  15  (1.0000, 3.8730)
  14 3 3 3  243  3  2  27  (1.5000, 2.5981)
  ...
  28 3 4 5  576  3  1  16  (3.0000, 4.0000)

Cf. A331248, A331249, A331251, A331252, A331253, A331695, A331696, A331697.

A331248 Common denominator of x-coordinate and y-coordinate of 3rd point of triangles with integer sides corresponding to A331247 and A331249.

Original entry on oeis.org

2, 2, 2, 1, 2, 4, 2, 1, 4, 2, 2, 4, 1, 2, 2, 6, 3, 2, 4, 1, 2, 6, 2, 4, 2, 1, 2, 1, 4, 2, 2, 8, 1, 1, 2, 2, 4, 3, 2, 6, 8, 1, 2, 2, 4, 8, 2, 2, 6, 1, 3, 2, 4, 1, 2, 8, 1, 10, 4, 2, 8, 1, 2, 2, 1, 2, 4, 2, 8, 1, 2, 6, 1, 10, 2, 2, 4, 2, 3, 1, 8, 2, 5, 5, 6, 8, 2

Offset: 1

Hugo Pfoertner, Jan 23 2020

			See A331247.

Cf. A331247, A331249, A331251, A331252, A331253.

A331249 Squared numerator of y-coordinate of 3rd point of n-th triangle with integer sides in a sorted list corresponding to A331247. The y-coordinate is given by sqrt(a(n))/A331248(n).

Original entry on oeis.org

3, 15, 35, 3, 63, 63, 99, 8, 135, 143, 195, 231, 15, 27, 255, 275, 80, 323, 351, 24, 399, 455, 483, 495, 55, 35, 575, 16, 663, 675, 75, 735, 48, 12, 783, 91, 855, 224, 899, 935, 975, 63, 63, 1023, 1071, 1071, 1155, 135, 1235, 80, 320, 1295, 1311, 21, 1443, 1463

Offset: 1

Hugo Pfoertner, Jan 23 2020

nonn

			a(1) = 3 because the 3rd point of the smallest triangle with integer sides (1,1,1) is at (x,y)=(1/2,sqrt(3)/2), A331247(1)=1, A331248(1)=2.
See A331247 for an extended list.

Cf. A331247, A331248, A331251, A331252, A331253, A331695, A331696, A331697.

cp's OEIS Frontend

A331251 Triangles with integer sides i <= j <= k sorted by area, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives shortest side i. The other sides are in A331252 and A331253.

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A331252 Triangles with integer sides i <= j <= k sorted by area, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives middle side j. The other sides are in A331251 and A331253.

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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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Mathematica

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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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A331250 a(n) = number of triangles with integer sides i <= j <= k with area <= n.

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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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A331248 Common denominator of x-coordinate and y-coordinate of 3rd point of triangles with integer sides corresponding to A331247 and A331249.

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A331249 Squared numerator of y-coordinate of 3rd point of n-th triangle with integer sides in a sorted list corresponding to A331247. The y-coordinate is given by sqrt(a(n))/A331248(n).

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