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

Author

Andrés Sancho, May 20 2024

Keywords

• nonn

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

Examples

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

Links

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

Crossrefs

Cf. A000292, A123323, A373041.

Programs

PARI

\\ See PARI link

Formula

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