A316853 -id:A316853 - 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

A055595 Area of triangles with integer sides and positive integer area, ordered by longest side, then second longest side and finally shortest side.

Original entry on oeis.org

6, 12, 12, 24, 48, 30, 60, 54, 24, 84, 48, 36, 60, 120, 108, 66, 42, 96, 84, 126, 60, 108, 192, 90, 150, 84, 168, 120, 36, 204, 240, 210, 210, 60, 120, 216, 132, 300, 96, 336, 72, 192, 144, 240, 480, 294, 84, 252, 360, 432, 114, 156, 180, 210, 420, 120, 210, 420

Offset: 1

Henry Bottomley, May 26 2000

nonn

This is the ordering of triangles used for A316841.

Ray Chandler, Table of n, a(n) for n = 1..10000
A. Dendane, Heron's Formula For Area of a Triangle-Geometry Calculator

The sides are given by A055592, A055593, A055594.
Cf. A046131, A070786, A316841, A316853.
Range of values: A188158.

max = 42; triangles = Reap[Do[s = (a+b+c)/2; area = Sqrt[s*(s-a)*(s-b)*(s-c)]; If[IntegerQ[area] && area > 0, Sow[{a, b, c, area}]], {a, 1, max}, {b, a, max}, {c, b, max}]][[2, 1]]; A055595 = Sort[triangles, #1[[3]]*max^2 + #1[[2]]*max + #1[[1]] < #2[[3]]* max^2 + #2[[2]]*max + #2[[1]] &][[All, 4]](* Jean-François Alcover, Jun 12 2012 *)

a(n) = sqrt(s(n)*(s(n)-A055592(n))*(s(n)-A055593(n))*(s(n)-A055594(n))) where s(n) = (A055592(n)+A055593(n)+A055594(n))/2 i.e. half the perimeter of the triangle

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).

A046131 Areas of scalene integer Heronian triangles (A046128, A046129, A046130) sorted by increasing c and b.

Original entry on oeis.org

6, 24, 30, 54, 24, 84, 36, 60, 66, 42, 96, 84, 126, 90, 150, 84, 120, 36, 204, 210, 210, 60, 216, 132, 96, 336, 72, 144, 240, 294, 84, 252, 360, 114, 156, 180, 210, 120, 210, 420, 168, 270, 264, 168, 384, 240, 468, 126, 180, 336, 336, 504, 264, 330, 486, 216

Offset: 0

Eric W. Weisstein

nonn

This is the ordering of triangles used for A316841.

Eric Weisstein's World of Mathematics, Heronian Triangle.

The sides are given by A046128, A046129, A046130.
Cf. A055595, A072294, A316841, A316853.
Range of values: A383413.

sideMax = 60; r[c_] := Reap[Do[ p = (a + b + c)/2; red = Reduce[ area > 1 && a < b < c && area^2 == p*(p - a)*(p - b)*(p - c), area, Integers]; If[red =!= False, sol = {a, b, c, area} /. {ToRules[red]}; Sow[sol]], {b, 1, c - 1}, {a, c - b, b - 1}]]; triangles = Flatten[ Reap[ Do[rc = r[c]; If[rc[[2]] =!= {}, Sow[rc[[2, 1]]]], {c, 5, sideMax}]][[2, 1]] , 2]; Sort[ triangles, Which[#1[[3]] < #2[[3]], True, #1[[3]] > #2[[3]], False, #1[[2]] < #2[[2]], True,  #1[[2]] > #2[[2]], False, #1[[1]] <= #2[[1]], True, True, False] &][[All, 4]] (* Jean-François Alcover, Oct 29 2012 *)

A317182 Numbers k such that k = 16*area(T)^2 for an integer triangle, T.

Original entry on oeis.org

3, 15, 35, 48, 63, 99, 128, 135, 143, 195, 231, 240, 243, 255, 275, 320, 323, 351, 384, 399, 455, 483, 495, 560, 575, 576, 663, 675, 735, 768, 783, 819, 855, 896, 899, 935, 975, 1008, 1023, 1071, 1155, 1215, 1235, 1280, 1295, 1311, 1344, 1443, 1463, 1536, 1539

Offset: 1

N. J. A. Sloane, Jul 25 2018

nonn

The possible lengths of the chord connecting the cusps of the lens-like intersection area between two circles with integer radii and integer distance d between their centers are of the form sqrt(a(n))/d. - Hugo Pfoertner, Sep 05 2020

Lars Blomberg, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Circle-Circle Intersection

Cf. A316841.

Sorted and uniqued values of A135622 or A316853.

More terms from Lars Blomberg, Apr 25 2019

Name edited by Peter Munn, Jul 30 2025

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.

A317181 Consider primitive integer triangles as listed in rows of table A316842. Sequence gives 16*area^2 for these triangles in the same order.

Original entry on oeis.org

3, 15, 63, 35, 128, 135, 320, 63, 495, 275, 231, 576, 975, 99, 384, 819, 1344, 455, 351, 896, 1575, 2304, 143, 2975, 735, 675, 1536, 2499, 495, 1280, 2295, 3456, 4655, 195, 768, 1683, 2880, 4275, 5760, 1071, 2304, 935, 3591, 663, 1728, 3135, 4800, 6615, 8448, 255, 2223, 5775, 10143

Offset: 1

N. J. A. Sloane, Jul 23 2018

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

Cf. A316842, A316852, A316846, A316847, A316848.

Subsequence of A316853.

a(n) = 16*s*(s-a)*(s-b)*(s-c) where s = A316852(n)/2, and a,b,c are A316843(n), A316844(n), A316845(n).

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.

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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A055595 Area of triangles with integer sides and positive integer area, ordered by longest side, then second longest side and finally shortest side.

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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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A046131 Areas of scalene integer Heronian triangles (A046128, A046129, A046130) sorted by increasing c and b.

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Mathematica

A317182 Numbers k such that k = 16*area(T)^2 for an integer triangle, T.

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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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A317181 Consider primitive integer triangles as listed in rows of table A316842. Sequence gives 16*area^2 for these triangles in the same order.

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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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