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

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

1,1

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Formula

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.

Example

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.

Prog

(Python)
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

Crossrefs

For integer triangles listed by area, see A331251, A331252, A331253.

For lists indicating the precise area of every integer triangle, see A135622 (which has further crossrefs) and A316853; also the related information in A317182 and A331011.

Sequence in context: A276211 A190091 A343148 * A186783 A133931 A050895

Adjacent sequences: A331247 A331248 A331249 * A331251 A331252 A331253

Keyword

nonn

Author

Hugo Pfoertner, Jan 20 2020

Status

approved