A317182 -id:A317182 - cp's OEIS frontend

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

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

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).

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.

A135622 16*Area^2 of integer triangles [A070080(n),A070081(n),A070082(n)].

Original entry on oeis.org

3, 15, 48, 35, 63, 128, 63, 135, 243, 240, 320, 99, 231, 275, 495, 384, 576, 768, 143, 351, 455, 819, 975, 560, 896, 1008, 1344, 195, 495, 675, 1215, 735, 1575, 1875, 768, 1280, 1536, 2048, 2304, 255, 663, 935, 1683, 1071, 2295, 2499, 2975, 1008, 1728

Offset: 1

Franz Vrabec, Feb 29 2008

			A070080(4)=1, A070081(4)=3, A070082(4)=3, so a(4)=(1+3+3)*(-1+3+3)*(1-3+3)*(1+3-3)=35.

See the formula section for the relationships with A070080, A070081, A070082, A070086.
Cf. A317182 (range of values), A331011 (nonunique values), A331250 (counts triangles by area).
Cf. A316853 (with terms ordered as for A316841), and using this order for other sets of triangles: A046131, A055595, A070786.

a(n)=(u+v+w)*(-u+v+w)*(u-v+w)*(u+v-w), where u=A070080(n), v=A070081(n), w=A070082(n).

A070086(n) = round(sqrt(a(n))/4).

A331011 16 * squared area of triangles with integer sides i <= j <= k, such that more than one ordered triple of sides produces the same area.

Original entry on oeis.org

63, 495, 675, 768, 1008, 1071, 1280, 1575, 2304, 2499, 2835, 2880, 2975, 3135, 3456, 3591, 4095, 4275, 4455, 4608, 4928, 5103, 5760, 5775, 6615, 6656, 6831, 6975, 7040, 7488, 7875, 7920, 8064, 8415, 8448, 8463, 8775, 8855, 8960, 9135, 9216, 9600, 9984, 10535

Offset: 1

Hugo Pfoertner, Jan 06 2020

nonn

			a(1) = 63: triangles (1,4,4) and (2,2,3) both have squared area 3.9375 = 63/16. All smaller squared area values A(i,j,k) correspond to unique triples of side lengths: A(1,1,1) = 0.1875 = 3/16, A(1,2,2) = 0.9375 = 15/16, A(1,3,3) = 2.1875 = 35/16, A(2,2,2) = 3 = 48/16.
a(2) = 495: A(2,6,7) = A(3,4,4) = 30.9375 = 495/16.
a(8) = 1575: A(2,11,12) = A(3,8,10) = A(4,5,6) = 98.4375 = 1575/16.
a(23) = 5760: A(2,19,19) = A(3,13,14) = A(6,7,7) = A(6,7,11) = 360 = 5760/16.

Subset of A317182.
Nonunique terms of A135622 or A316853.
Cf. A331012.

A331012 16 * squared area of triangles with integer sides i <= j <= k, such that more triples of sides produce the same area as for any smaller area.

Original entry on oeis.org

3, 63, 1575, 5760, 24255, 51975, 80640, 172800, 322560, 403200, 1209600, 2822400, 4435200, 8870400, 15523200, 17740800, 53222400, 125798400, 146764800

Offset: 1

Hugo Pfoertner, Jan 07 2020

The corresponding record counts of triangles are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 20, 26, 30, 34, 35, ... .

Conjectured further terms are 226195200 (42 representations), 230630400 (52 representations), 372556800 (55 representations).

			See A331011.

Cf. A316841, A316853, A317182, A331011.

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

Hugo Pfoertner, Jan 12 2020

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.

			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).

Cf. A317182, A331223, A331227, A331228.

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.

A371973 a(n) is the number of distinct areas > 0 of triangles with integer sides and perimeter n.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 13, 19, 14, 21, 19, 23, 20, 27, 23, 30, 27, 32, 29, 35, 32, 39, 34, 44, 39, 48, 43, 52, 47, 55, 51, 60, 53, 63, 59, 69, 58, 74, 67, 78, 73, 84, 75, 90, 81, 92, 88, 101, 91, 108, 93, 112, 106

Offset: 3

Hugo Pfoertner, Apr 16 2024

nonn

Hugo Pfoertner, Table of n, a(n) for n = 3..10000

Cf. A005044, A007237, A024153, A317182, A371070.

See the formula section for the relationships with A026810, A070083, A135622 (which has many crossrefs related to areas of triangles).

A2(a,b,c) = {my (s=(a+b+c)/2); s*(s-a)*(s-b)*(s-c)};
a371973(n) = {my (A=List()); forpart (v=n, listput(A, A2(v[1],v[2],v[3])), [1,(n-1)\2], [3,3]); #Set(A)};

def A371973(n): return len(set((2*(b+c)-n)*(n-2*b)*(n-2*c) for c in range((n+2)//3, (n+1)//2) for b in range((n-c+1)//2, c+1))) # David Radcliffe, Aug 01 2025

a(n) = |{A135622(k) : A070083(k) = n}| = |{A135622(k) : A026810(n) < k <= A026810(n+1)}|. - Peter Munn, Jul 29 2025

b-file corrected by David Radcliffe, Aug 01 2025

A317183 Numbers k such that k = 16*area(T)^2 for a primitive integer triangle, T.

Original entry on oeis.org

3, 15, 35, 63, 99, 128, 135, 143, 195, 231, 255, 275, 320, 323, 351, 384, 399, 455, 483, 495, 575, 576, 663, 675, 735, 768, 783, 819, 855, 896, 899, 935, 975, 1023, 1071, 1155, 1235, 1280, 1295, 1311, 1344, 1443, 1463, 1536, 1539, 1575, 1599, 1683, 1728, 1763

Offset: 1

N. J. A. Sloane, Jul 25 2018

nonn

A316842 lists the primitive integer triangles.

Lars Blomberg, Table of n, a(n) for n = 1..10000

Cf. A316842.
Sorted and uniqued values of A317181.
Subsequence of A317182.

Terms a(13) and beyond from Lars Blomberg, Jul 27 2018

Name edited by Peter Munn, Jul 30 2025

cp's OEIS Frontend

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

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A135622 16*Area^2 of integer triangles [A070080(n),A070081(n),A070082(n)].

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A331011 16 * squared area of triangles with integer sides i <= j <= k, such that more than one ordered triple of sides produces the same area.

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A331012 16 * squared area of triangles with integer sides i <= j <= k, such that more triples of sides produce the same area as for any smaller area.

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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.

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A371973 a(n) is the number of distinct areas > 0 of triangles with integer sides and perimeter n.

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Python

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A317183 Numbers k such that k = 16*area(T)^2 for a primitive integer triangle, T.

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