A331011 -id:A331011 - 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).

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).

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

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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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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