A316841 -id:A316841 - cp's OEIS frontend

A316843 Column 1 of table A316841.

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

A316844 Column 2 of table A316841.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jul 23 2018

nonn

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

Cf. A316841, A316843, A316845.

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

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).

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).

Original entry on oeis.org

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.

A331257 Numerator of squared radius of inscribed circle of the n-th triangle with integer sides in the list given by A316841. Denominators are A331258.

Original entry on oeis.org

1, 3, 1, 9, 5, 1, 3, 5, 4, 7, 3, 45, 4, 25, 21, 1, 75, 9, 2, 63, 12, 25, 35, 9, 27, 8, 7, 9, 11, 5, 27, 2, 175, 3, 49, 3, 3, 147, 11, 5, 135, 8, 245, 13, 3, 99, 20, 225, 18, 49, 63, 16, 55, 5, 189, 16, 39, 4, 165, 3, 15, 48, 15, 7, 117, 12, 275, 45, 441, 16

Offset: 1

Hugo Pfoertner, Jan 26 2020

			   n i (A316843)
   | | j (A316844)
   | | | k (A316845)
   | | | |  a(n) this sequence
   | | | |  |  A331258(n)
   | | | |  |  |  rho = sqrt(a(n)/A331258(n))
   1 1 1 1  1 12  0.28868
   2 2 2 1  3 20  0.38730
   3 2 2 2  1  3  0.57735
   4 3 2 2  9 28  0.56695
   5 3 3 1  5 28  0.42258
   6 3 3 2  1  2  0.70711
   7 3 3 3  3  4  0.86603
   8 4 3 2  5 12  0.64550
   9 4 3 3  4  5  0.89443
  10 4 4 1  7 36  0.44096
  11 4 4 2  3  5  0.77460
  12 4 4 3 45 44  1.01130
  13 4 4 4  4  3  1.15470
  14 5 3 3 25 44  0.75378
  15 5 4 2 21 44  0.69085
  16 5 4 3  1  1  1.00000

Cf. A316841, A316843, A316844, A316845, A331258.

rh2(a, b, c)={my(s=(a+b+c)/2); (s-a)*(s-b)*(s-c)/s};
for(i=1, 8, for(j=1, i, for(k=1, j, if(j+k>i, print1(numerator(rh2(i, j, k)), ", ")))))

Radius of inscribed circle of a triangle with sides (a,b,c):

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

A331258 Denominator of squared radius of inscribed circle of the n-th triangle with integer sides in the list given by A316841. Numerators are A331257.

Original entry on oeis.org

12, 20, 3, 28, 28, 2, 4, 12, 5, 36, 5, 44, 3, 44, 44, 1, 52, 44, 3, 52, 7, 12, 52, 7, 52, 7, 4, 4, 52, 7, 20, 1, 68, 1, 60, 4, 2, 68, 20, 4, 68, 3, 76, 60, 4, 68, 9, 76, 5, 12, 68, 9, 68, 3, 76, 5, 68, 3, 76, 1, 4, 11, 68, 9, 76, 5, 84, 11, 92, 3, 76, 76, 1

Offset: 1

Hugo Pfoertner, Jan 26 2020

			See A331257.

Cf. A316841, A316843, A316844, A316845, A331257.

rh2(a,b,c)={my(s=(a+b+c)/2); (s-a)*(s-b)*(s-c)/s};
for(i=1,8, for(j=1,i, for(k=1,j, if(j+k>i, print1(denominator(rh2(i,j,k)),", ")))))

A316849 The table in A316841 with columns concatenated to form a single number.

Original entry on oeis.org

111, 221, 222, 322, 331, 332, 333, 432, 433, 441, 442, 443, 444, 533, 542, 543, 544, 551, 552, 553, 554, 555, 643, 644, 652, 653, 654, 655, 661, 662, 663, 664, 665, 666, 744, 753, 754, 755, 762, 763, 764, 765, 766, 771, 772, 773, 774, 775, 776, 777, 854, 855

Offset: 1

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

nonn

This makes no sense once the entries in A316841 exceed 9, but is included because some people may search for this version. See A316841 for the official version.

Does NOT need a b-file.

Cf. A316841.

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).

cp's OEIS Frontend

A316843 Column 1 of table A316841.

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A316844 Column 2 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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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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A331257 Numerator of squared radius of inscribed circle of the n-th triangle with integer sides in the list given by A316841. Denominators are A331258.

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A331258 Denominator of squared radius of inscribed circle of the n-th triangle with integer sides in the list given by A316841. Numerators are A331257.

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Examples

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PARI

A316849 The table in A316841 with columns concatenated to form a single number.

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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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