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A375815 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 A375815 #8 Aug 31 2024 08:32:33
%S A375815 1,2,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,
%T A375815 10,10,10,11,10,11,11,11,11,11,12,12,12,12,12,12,12,12,13,12,13,13,13,
%U A375815 13,13,13,14,14,14,14,14,14,14,14,14,15,14,15,15,15
%N A375815 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 A375815 The variant with strict inequality (A375814) is finite; is this sequence infinite?
%H A375815 Rémy Sigrist, <a href="/A375815/b375815.txt">Table of n, a(n) for n = 1..10000</a>
%e A375815 The first terms, alongside the corresponding sums, are:
%e A375815   n   a(n)  Sum {k=1..n} 1/(a(k)*a(n+1-k))
%e A375815   --  ----  ------------------------------
%e A375815    1     1  1
%e A375815    2     2  1
%e A375815    3     3  11/12
%e A375815    4     3  1
%e A375815    5     4  17/18
%e A375815    6     4  35/36
%e A375815    7     5  167/180
%e A375815    8     5  14/15
%e A375815    9     5  77/80
%e A375815   10     5  119/120
%e A375815   11     6  77/80
%e A375815   12     6  29/30
%e A375815   13     6  443/450
%e A375815   14     7  3007/3150
%e A375815   15     7  6011/6300
%o A375815 (PARI) { for (n = 1, #a = vector(72), if (n==1, a[n] = 1, x = sum(k = 2, n-1, 1/(a[k]*a[n+1-k])); if (x >= 1, break, a[n] = ceil(2/(a[1]*(1-x))););); print1 (a[n]", ");); }
%Y A375815 Cf. A375814, A375834.
%K A375815 nonn
%O A375815 1,2
%A A375815 _Rémy Sigrist_, Aug 30 2024