A375423 a(1) = 1; for any n > 1, a(n) is the maximum number of points from the set {(k, a(k)), k = 1..n-1} belonging to a straight line passing through the point (n-1, a(n-1)).

1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 3, 5, 5, 4, 4, 5, 3, 6, 4, 6, 3, 7, 5, 4, 7, 4, 8, 4, 9, 5, 5, 6, 4, 10, 6, 4, 11, 6, 5, 7, 3, 8, 3, 9, 4, 12, 3, 10, 4, 13, 3, 11, 5, 8, 3, 12, 6, 6, 7, 4, 14, 4, 15, 4, 16, 4, 17, 4, 18, 5, 9, 4, 19, 4, 20, 5, 10, 3, 13, 3

Offset: 1

Author

Rémy Sigrist, Aug 14 2024

Keywords

• nonn

Comments

This sequence is unbounded (if the sequence was bounded, say by m, then, by the pigeonhole principle, some value v <= m would appear infinitely many times, and for any k > 0, the k-th occurrence of v would be followed by a value >= k, a contradiction).

Examples

			The first terms, alongside an appropriate set of points, are:
  n   a(n)  Points
  --  ----  -----------------------------------
   1     1  N/A
   2     1  (1,1)
   3     2  (1,1), (2,1)
   4     2  (1,1), (3,2)
   5     2  (1,1), (4,2)
   6     3  (3,2), (4,2), (5,2)
   7     3  (2,1), (4,2), (6,3)
   8     3  (1,1), (4,2), (7,3)
   9     3  (2,1), (5,2), (8,3)
  10     4  (6,3), (7,3), (8,3), (9,3)
  11     4  (1,1), (4,2), (7,3), (10,4)
  12     4  (2,1), (5,2), (8,3), (11,4)
  13     3  (4,2), (8,3), (12,4)
  14     5  (6,3), (7,3), (8,3), (9,3), (13,3)
  15     5  (2,1), (5,2), (8,3), (11,4), (14,5)
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, C++ program
• Rémy Sigrist, PARI program

Crossrefs

Cf. A334043, A375422.

Programs

PARI

\\ See Links section.