A373051 Number of non-similar triangles possible with distinct positive integer side lengths of at most n units.

0, 0, 0, 1, 3, 7, 13, 21, 33, 47, 67, 87, 117, 147, 187, 227, 283, 331, 403, 467, 551, 631, 741, 829, 959, 1073, 1217, 1349, 1531, 1667, 1877, 2053, 2273, 2473, 2737, 2941, 3247, 3499, 3811, 4083, 4463, 4739, 5159, 5499, 5907, 6281, 6787, 7155, 7701, 8131, 8675, 9155, 9805

Offset

1,5

Comments

Also, number of triangles possible with integer side lengths x, y, and z such that z < y < x <= n and gcd(x, y, z) = 1.

For all n, this number is strictly less than n^3. For all n > 5, this number is strictly greater than n.

For all n > 3, this sequence is strictly increasing.

The first n terms can be calculated in O(n^3) time.

a(n) <= A000292(n + 2). - David A. Corneth, May 22 2024

Links

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 2893 terms from Andrés Sancho)

David A. Corneth, PARI program

Formula

a(n) = 1 + 2*Sum_{k=5..n} A373041(k) for n >= 5.

Example

For n = 5, the 3 solutions are (4, 3, 2), (5, 4, 2), and (5, 4, 3).

Prog

(C++) int a(int n) { int term = 0; for (int x = 4; x <= n; x++) { for (int y = ceil(x / 2.0f) + 1; y < x; y++) { for (int z = x - y + 1; z < y; z++) { if (gcd(gcd(x, y), z) == 1) { term++; } } } } return term; }
(PARI) \\ See PARI link

Crossrefs

Cf. A000292, A123323, A373041.

Sequence in context: A172310 A357370 A060939 * A174030 A355734 A098575

Adjacent sequences: A373045 A373049 A373050 * A373052 A373053 A373054

Keyword

nonn,changed

Author

Andrés Sancho, May 20 2024

Status

approved