A385807 - OEIS

%I #42 Jul 21 2025 03:05:50

%S 0,1,3,6,10,13,18,23,27,32,39,42,50,55,60,67,74,79,87,92,99,106,115,

%T 118,128,135,140,149,158,161,172,179,187,194,201,208,219,226,233,240,

%U 252,255,268,273,280,293,300,305,316,325,333,340,353,356,367,376,385,394,403,408,424,429

%N 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.

%C The count excludes points on the base and edges.

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

%e 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.

%o (Python)

%o 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))

%o print([a(n) for n in range(1, 63)])

%o (PARI) a(n) = sum(x=1, 2*n-1, sumdiv(x, y, y < min(x, 2*n-x))); \\ _Michel Marcus_, Jul 11 2025

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

%K nonn

%O 1,3

%A _Rickey W. Austin_, Jul 09 2025