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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.
2

%I #11 Jun 18 2019 12:24:25

%S 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,

%T 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,

%U 45,0,56,0,69,1,56,3,16,4,5,3,25,1,69,1,79,0,82,1

%N 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.

%C This sequence is unbounded: by contradiction:

%C - 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), ...),

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

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

%H Rémy Sigrist, <a href="/A308639/b308639.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A308639/a308639.txt">C program for A308639</a>

%e 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:

%e n a(n) (i,j)'s

%e -- ---- -----------------------------------

%e 1 0 none

%e 2 0 none

%e 3 0 none

%e 4 1 (1,2)

%e 5 0 none

%e 6 3 (1,2), (1,3), (2,3)

%e 7 1 (3,4)

%e 8 0 none

%e 9 6 (1,2), (1,3), (1,5), (2,3), (2,5), (3,5)

%e 10 3 (3,4), (3,6), (4,6)

%e 11 1 (1,4)

%e 12 1 (4,7)

%e 13 3 (4,7), (4,11), (7,11)

%e 14 1 (6,10)

%o (C) See Links section.

%Y See A308638 for a similar sequence.

%K nonn

%O 1,6

%A _Rémy Sigrist_, Jun 13 2019