A316844 -id:A316844 - cp's OEIS frontend

A331695 Numerator of the x-coordinate of the 3rd point (x,y) of the n-th triangle with integer sides in the list given by A316841, when the triangle is drawn with the longest side from (0,0) to (0,A316843(n)) and the middle side A316844(n) from (0,A316843(n)) to (x,y). x = a(n)/A331696(n), y = sqrt(A331697(n))/A331696(n).

1, 1, 1, 3, 1, 2, 3, 11, 2, 1, 1, 9, 2, 5, 13, 9, 5, 1, 2, 9, 8, 5, 29, 3, 5, 5, 9, 3, 1, 1, 3, 4, 25, 3, 7, 33, 20, 7, 17, 11, 29, 19, 7, 1, 2, 9, 8, 25, 18, 7, 55, 4, 37, 11, 53, 4, 19, 3, 31, 5, 51, 4, 1, 1, 9, 1, 25, 9, 49, 4, 9, 61, 35, 9, 41, 8, 19, 34

Offset: 1

Hugo Pfoertner, Jan 25 2020

The shortest side of the triangle has length A316845(n), i.e., x^2 + y^2 = A316845(n)^2.

			x(n) = a(n)/A331696(n),
y(n) = sqrt(A331697(n))/A331696(n).
   n i (A316843)
   | | j (A316844)
   | | | k (A316845)
   | | | |  a(n) this sequence
   | | | |  |  A331696
   | | | |  |  |   A331697
   | | | |  |  |   |  (x,y)
   1 1 1 1  1  2   3  (0.5000,0.86603)
   2 2 2 1  1  4  15  (0.2500,0.96825)
   3 2 2 2  1  1   3  (1.0000,1.7321)
   4 3 2 2  3  2   7  (1.5000,1.3229)
   5 3 3 1  1  6  35  (0.16667,0.98601)
   6 3 3 2  2  3  32  (0.66667,1.8856)
   7 3 3 3  3  2  27  (1.5000,2.5981)
   8 4 3 2 11  8 135  (1.3750,1.4524)
   9 4 3 3  2  1   5  (2.0000,2.2361)
  10 4 4 1  1  8  63  (0.12500,0.99216)
  11 4 4 2  1  2  15  (0.50000,1.9365)
  12 4 4 3  9  8 495  (1.1250,2.7811)
  13 4 4 4  2  1  12  (2.0000,3.4641)
  14 5 3 3  5  2  11  (2.5000,1.6583)
  15 5 4 2 13 10 231  (1.3000,1.5199)
  16 5 4 3  9  5 144  (1.8000,2.4000)

Cf. A316841.
Sides of triangle: A316843, A316844, A316845.
Cf. A331696, A331697.

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).

A316841 Three-column table read by rows giving integer sides of proper triangles (i,j,k) with i >= j >= k >= 1, j+k > i.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jul 23 2018, following a suggestion from Donald S. McDonald

			Table begins (imprimitive triples are labeled i):
[1,1,1],
[2,2,1],
[2,2,2],i
[3,2,2],
[3,3,1],
[3,3,2],
[3,3,3],i
[4,3,2],
[4,3,3],
[4,4,1],
[4,4,2],i
[4,4,3],
[4,4,4],i
[5,3,3],
...

Lars Blomberg, Table of n, a(n) for n = 1..9000

There are A002620(k+1) rows that begin with k.
The three columns are A316843, A316844, A316845.
A316849 is a compressed version.
See A316842 for primitive triples.
See A316851 and A316853 & A317182 for perimeter and area.
Other related sequences: A051493, A070080, A070081, A070082, A070110.

for(i=1,6, for(j=1,i, for(k=1,j, if(j+k>i, print1(i,", ",j,", ",k,", "))))) \\ Hugo Pfoertner, Jan 25 2020

A316843 Column 1 of table A316841.

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, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 1

N. J. A. Sloane, Jul 23 2018

nonn

Lars Blomberg, Table of n, a(n) for n = 1..3000

Cf. A002620, A316841, A316844, A316845.

k appears A002620(k+1) times.

More terms from Lars Blomberg, Apr 25 2019

A316845 Column 3 of table A316841.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jul 23 2018

nonn

Lars Blomberg, Table of n, a(n) for n = 1..3000

Cf. A316841, A316843, A316844.

More terms from Lars Blomberg, Apr 25 2019

A316853 Areas of all nondegenerate integer triangles, T, expressed as 16*area(T)^2. a(n) is for the triangle with sides A316841(n, 1..3).

Original entry on oeis.org

3, 15, 48, 63, 35, 128, 243, 135, 320, 63, 240, 495, 768, 275, 231, 576, 975, 99, 384, 819, 1344, 1875, 455, 1008, 351, 896, 1575, 2304, 143, 560, 1215, 2048, 2975, 3888, 735, 675, 1536, 2499, 495, 1280, 2295, 3456, 4655, 195, 768, 1683, 2880, 4275

Offset: 1

N. J. A. Sloane, Jul 23 2018

nonn

The squared area of an integer triangle is necessarily a multiple of 1/16.

Lars Blomberg, Table of n, a(n) for n = 1..3000
Eric Weisstein's World of Mathematics, Triangle Area.

Cf. A316841, A316842, A316843, A316844, A316845, A316851.

a(n) = 16*s*(s-a)*(s-b)*(s-c) where s = A316851(n)/2, and a,b,c are A316843(n), A316844(n), A316845(n).

Name edited by Peter Munn, May 10 2025

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.

Original entry on oeis.org

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).

A316851 Consider integer triangles as listed in rows of table A316841. Sequence gives perimeters of these triangles in the same order.

Original entry on oeis.org

3, 5, 6, 7, 7, 8, 9, 9, 10, 9, 10, 11, 12, 11, 11, 12, 13, 11, 12, 13, 14, 15, 13, 14, 13, 14, 15, 16, 13, 14, 15, 16, 17, 18, 15, 15, 16, 17, 15, 16, 17, 18, 19, 15, 16, 17, 18, 19, 20, 21, 17, 18, 17, 18, 19, 20, 17, 18, 19, 20, 21, 22, 17, 18, 19, 20, 21, 22, 23, 24

Offset: 1

N. J. A. Sloane, Jul 23 2018

Every number appears except 1, 2, and 4.

Lars Blomberg, Table of n, a(n) for n = 1..3000

Cf. A316841, A316842, A316852.

a(n) = A316843(n)+A316844(n)+A316845(n).

A331696 Common denominator of x-coordinate and y-coordinate of 3rd point of triangles with integer sides corresponding to A331695 and A331697.

Original entry on oeis.org

2, 4, 1, 2, 6, 3, 2, 8, 1, 8, 2, 8, 1, 2, 10, 5, 2, 10, 5, 10, 5, 2, 12, 1, 4, 3, 4, 1, 12, 3, 4, 3, 12, 1, 2, 14, 7, 2, 14, 7, 14, 7, 2, 14, 7, 14, 7, 14, 7, 2, 16, 1, 16, 4, 16, 1, 16, 2, 16, 2, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 2, 18, 9, 2, 18, 3, 6, 9, 2

Offset: 1

Hugo Pfoertner, Jan 25 2020

			See A331695.

Cf. A316841.
Sides of triangle: A316843, A316844, A316845.
Cf. A331695, A331697.

A331697 Squared numerator of y-coordinate of 3rd point of the n-th triangle with integer sides in a sorted list corresponding to A331695. The y-coordinate is given by sqrt(a(n))/A331696(n).

Original entry on oeis.org

3, 15, 3, 7, 35, 32, 27, 135, 5, 63, 15, 495, 12, 11, 231, 144, 39, 99, 96, 819, 336, 75, 455, 7, 39, 56, 175, 16, 143, 35, 135, 128, 2975, 27, 15, 675, 384, 51, 495, 320, 2295, 864, 95, 195, 192, 1683, 720, 4275, 1440, 147, 1071, 9, 935, 135, 3591, 20, 663, 27

Offset: 1

Hugo Pfoertner, Jan 25 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), A331695(1)=1, A331696(1)=2.
See A331695 for an extended list.

Cf. A316841.
Sides of triangle: A316843, A316844, A316845.
Cf. A331695, A331696.

cp's OEIS Frontend

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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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A316841 Three-column table read by rows giving integer sides of proper triangles (i,j,k) with i >= j >= k >= 1, j+k > i.

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A316843 Column 1 of table A316841.

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A316845 Column 3 of table A316841.

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A316853 Areas of all nondegenerate integer triangles, T, expressed as 16*area(T)^2. a(n) is for the triangle with sides A316841(n, 1..3).

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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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A316851 Consider integer triangles as listed in rows of table A316841. Sequence gives perimeters of these triangles in the same order.

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A331696 Common denominator of x-coordinate and y-coordinate of 3rd point of triangles with integer sides corresponding to A331695 and A331697.

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A331697 Squared numerator of y-coordinate of 3rd point of the n-th triangle with integer sides in a sorted list corresponding to A331695. The y-coordinate is given by sqrt(a(n))/A331696(n).

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