A385807 Number of integer lattice points (x, y) strictly inside a triangle of base 2n - 1 and height n - 1, such that 1 <= x <= 2n - 1, 1 <= y < min(x, 2n - x), and y | x.

0, 1, 3, 6, 10, 13, 18, 23, 27, 32, 39, 42, 50, 55, 60, 67, 74, 79, 87, 92, 99, 106, 115, 118, 128, 135, 140, 149, 158, 161, 172, 179, 187, 194, 201, 208, 219, 226, 233, 240, 252, 255, 268, 273, 280, 293, 300, 305, 316, 325, 333, 340, 353, 356, 367, 376, 385, 394, 403, 408, 424, 429

Offset

1,3

Comments

The count excludes points on the base and edges.

Links

Table of n, a(n) for n=1..62.

Formula

a(n) = |{ (x, y) : 1 <= x <= 2n - 1, 1 <= y < min(x, 2n - x), and y divides x }|.

Example

For n = 4, the triangle has x in [1,7]. Valid (x, y) points satisfying y < min(x, 8 - x) and y divides x are: (2,1), (3,1), (4,1), (4,2), (5,1). So a(4) = 5.

Prog

(Python)
def a(n): return sum(1 for y in range(1, n) for k in range(1, (2*n)//y + 1) if y < min(y*k, 2*n - y*k))
print([a(n) for n in range(1, 63)])
(PARI) a(n) = sum(x=1, 2*n-1, sumdiv(x, y, y < min(x, 2*n-x))); \\ Michel Marcus, Jul 11 2025

Crossrefs

Cf. A000005 (number of divisors), A032741 (divisors < n), A339217 (for optimized generation of lattice divisor patterns).

Sequence in context: A267593 A248221 A189368 * A128039 A027428 A382729

Adjacent sequences: A385804 A385805 A385806 * A385808 A385809 A385810

Keyword

nonn

Author

Rickey W. Austin, Jul 09 2025

Status

approved