This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A308639 #11 Jun 18 2019 12:24:25 %S A308639 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 A308639 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 A308639 45,0,56,0,69,1,56,3,16,4,5,3,25,1,69,1,79,0,82,1 %N 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. %C A308639 This sequence is unbounded: by contradiction: %C A308639 - 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 A308639 - hence for any k > 0, a(b(k)+1) >= (k-1)*(k-2)/2, %C A308639 - and for k > 2 + sqrt(2*M), a(b(n)+1) > M , a contradiction, QED. %H A308639 Rémy Sigrist, <a href="/A308639/b308639.txt">Table of n, a(n) for n = 1..10000</a> %H A308639 Rémy Sigrist, <a href="/A308639/a308639.txt">C program for A308639</a> %e A308639 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 A308639 n a(n) (i,j)'s %e A308639 -- ---- ----------------------------------- %e A308639 1 0 none %e A308639 2 0 none %e A308639 3 0 none %e A308639 4 1 (1,2) %e A308639 5 0 none %e A308639 6 3 (1,2), (1,3), (2,3) %e A308639 7 1 (3,4) %e A308639 8 0 none %e A308639 9 6 (1,2), (1,3), (1,5), (2,3), (2,5), (3,5) %e A308639 10 3 (3,4), (3,6), (4,6) %e A308639 11 1 (1,4) %e A308639 12 1 (4,7) %e A308639 13 3 (4,7), (4,11), (7,11) %e A308639 14 1 (6,10) %o A308639 (C) See Links section. %Y A308639 See A308638 for a similar sequence. %K A308639 nonn %O A308639 1,6 %A A308639 _Rémy Sigrist_, Jun 13 2019