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%I #15 Aug 17 2024 12:21:05
%S 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,
%T 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,
%U 15,4,16,4,17,4,18,5,9,4,19,4,20,5,10,3,13,3
%N 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)).
%C 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).
%H Rémy Sigrist, <a href="/A375423/b375423.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A375423/a375423_1.txt">C++ program</a>
%H Rémy Sigrist, <a href="/A375423/a375423.gp.txt">PARI program</a>
%e The first terms, alongside an appropriate set of points, are:
%e n a(n) Points
%e -- ---- -----------------------------------
%e 1 1 N/A
%e 2 1 (1,1)
%e 3 2 (1,1), (2,1)
%e 4 2 (1,1), (3,2)
%e 5 2 (1,1), (4,2)
%e 6 3 (3,2), (4,2), (5,2)
%e 7 3 (2,1), (4,2), (6,3)
%e 8 3 (1,1), (4,2), (7,3)
%e 9 3 (2,1), (5,2), (8,3)
%e 10 4 (6,3), (7,3), (8,3), (9,3)
%e 11 4 (1,1), (4,2), (7,3), (10,4)
%e 12 4 (2,1), (5,2), (8,3), (11,4)
%e 13 3 (4,2), (8,3), (12,4)
%e 14 5 (6,3), (7,3), (8,3), (9,3), (13,3)
%e 15 5 (2,1), (5,2), (8,3), (11,4), (14,5)
%o (C++) // See Links section.
%o (PARI) \\ See Links section.
%Y Cf. A334043, A375422.
%K nonn
%O 1,3
%A _Rémy Sigrist_, Aug 14 2024