A308639 a(n) is the number of pairs (i,j) such that 0 < i < j < n-1 and the points (i, a(i)), (j, a(j)) and (n-1, a(n-1)) are aligned.

0, 0, 0, 1, 0, 3, 1, 0, 6, 3, 1, 1, 3, 1, 6, 1, 10, 0, 11, 2, 2, 2, 2, 5, 0, 15, 1, 16, 0, 21, 1, 22, 2, 7, 1, 29, 2, 11, 0, 31, 2, 16, 1, 36, 9, 0, 38, 3, 5, 2, 21, 0, 45, 4, 3, 11, 3, 12, 1, 45, 0, 56, 0, 69, 1, 56, 3, 16, 4, 5, 3, 25, 1, 69, 1, 79, 0, 82, 1

Offset: 1

Author

Rémy Sigrist, Jun 13 2019

Keywords

• nonn

Comments

This sequence is unbounded: by contradiction:

- if the sequence was bounded, say a(n) <= M for any n > 0, then some value, say v, would appear infinitely many times, say at indices (b(1), b(2), ...),

- hence for any k > 0, a(b(k)+1) >= (k-1)*(k-2)/2,

- and for k > 2 + sqrt(2*M), a(b(n)+1) > M , a contradiction, QED.

Examples

			The first terms, alongside the pairs (i,j) such that 0 < i < j < n-1 and the points (i, a(i)), (j, a(j)) and (n-1, a(n-1)) are aligned, are:
  n   a(n)  (i,j)'s
  --  ----  -----------------------------------
   1     0  none
   2     0  none
   3     0  none
   4     1  (1,2)
   5     0  none
   6     3  (1,2), (1,3), (2,3)
   7     1  (3,4)
   8     0  none
   9     6  (1,2), (1,3), (1,5), (2,3), (2,5), (3,5)
  10     3  (3,4), (3,6), (4,6)
  11     1  (1,4)
  12     1  (4,7)
  13     3  (4,7), (4,11), (7,11)
  14     1  (6,10)
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, C program for A308639

Crossrefs

See A308638 for a similar sequence.

Programs

C

See Links section.