A334044 a(1) = 0, and for any n > 1, a(n) is the number of points of the set { (k, a(k)), k = 1..n-2 } that are not visible from the point (n-1, a(n-1)).

0, 0, 0, 1, 0, 2, 1, 0, 4, 1, 1, 2, 2, 2, 3, 2, 3, 5, 2, 6, 4, 3, 3, 4, 4, 7, 1, 5, 5, 2, 6, 5, 4, 6, 6, 9, 0, 6, 9, 3, 4, 7, 5, 8, 5, 6, 6, 10, 8, 10, 7, 7, 7, 5, 6, 11, 6, 11, 8, 6, 14, 8, 8, 10, 11, 9, 8, 15, 8, 12, 8, 12, 11, 6, 14, 8, 12, 14, 10, 13, 8

Offset: 1

Author

Rémy Sigrist, Apr 13 2020

Keywords

• nonn

Comments

For any i and k such that i < k: the point (i, a(i)) is visible from the point (k, a(k)) if there are no j such that i < j < k and the three points (i, a(i)), (j, a(j)), (k, a(k)) are aligned.

Examples

			For n = 6:
- we consider the following points:
      .    .    .    X    .
                   (4,1)
      X    X    X    .    X
   (1,0) (2,0) (3,0)    (5,0)
- the points (1,0) and (2,0) are not visible from (5,0)
  (as they are hidden by (3,0)),
- whereas the points (3,0) and (4,1) are visible from (5,0)
- hence a(6) = 2.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000

Crossrefs

See A334043 for a similar sequence.

Programs

PARI

g(z) = z/gcd(real(z), imag(z))
for (n=1, #a=vector(81), print1 (a[n] = max(0, n-2) - #Set(apply(k -> g((k+a[k]*I)-(n-1+a[n-1]*I)), [1..n-2]))", "))