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A375814 Lexicographically earliest sequence of positive integers such that for any n > 0, Sum_{k = 1..n} 1/(a(k)*a(n+1-k)) < 1.

%I A375814 #10 Aug 31 2024 08:32:28
%S A375814 2,2,2,3,3,4,4,4,5,5,5,6,6,6,7,6,7,8,7,8,8,7,9,8,9,9,8,10,9,9,10,10,
%T A375814 10,10,10,10,11,10,11,11,11,12,11,11,13,11,12,12,12,13,12,12,14,12,13,
%U A375814 14,12,14,14,13,14,13,14,15,13,15,15,13,16,14,14,16
%N A375814 Lexicographically earliest sequence of positive integers such that for any n > 0, Sum_{k = 1..n} 1/(a(k)*a(n+1-k)) < 1.
%C A375814 Sum_{k = 2..300} 1/(a(k)*a(302-k)) > 1, hence the sequence is finite with 300 terms.
%H A375814 Rémy Sigrist, <a href="/A375814/b375814.txt">Table of n, a(n) for n = 1..300</a>
%e A375814 The first terms, alongside the corresponding sums, are:
%e A375814   n   a(n)  Sum_{k=1..n} 1/(a(k)*a(n+1-k))
%e A375814   --  ----  ------------------------------
%e A375814    1     2  1/4
%e A375814    2     2  1/2
%e A375814    3     2  3/4
%e A375814    4     3  5/6
%e A375814    5     3  11/12
%e A375814    6     4  11/12
%e A375814    7     4  17/18
%e A375814    8     4  35/36
%e A375814    9     5  44/45
%e A375814   10     5  59/60
%e A375814   11     5  239/240
%e A375814   12     6  119/120
%e A375814   13     6  79/80
%e A375814   14     6  119/120
%e A375814   15     7  991/1008
%o A375814 (PARI) { for (n = 1, #a = vector(72), if (n==1, a[n] = 2, x = sum(k = 2, n-1, 1/(a[k]*a[n+1-k])); if (x >= 1, break, a[n] = floor(2/(a[1]*(1-x)))+1;);); print1 (a[n]", ");); }
%Y A375814 Cf. A375815, A375834.
%K A375814 nonn,fini,full
%O A375814 1,1
%A A375814 _Rémy Sigrist_, Aug 30 2024