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

1,6

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.

Links

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

Rémy Sigrist, C program for A308639

Example

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)

Prog

(C) See Links section.

Crossrefs

See A308638 for a similar sequence.

Sequence in context: A371995 A071417 A096653 * A348191 A213668 A256595

Adjacent sequences: A308636 A308637 A308638 * A308640 A308641 A308642

Keyword

nonn

Author

Rémy Sigrist, Jun 13 2019

Status

approved