A375815 - OEIS

%I #8 Aug 31 2024 08:32:33

%S 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 10,10,10,11,10,11,11,11,11,11,12,12,12,12,12,12,12,12,13,12,13,13,13,

%U 13,13,13,14,14,14,14,14,14,14,14,14,15,14,15,15,15

%N 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 The variant with strict inequality (A375814) is finite; is this sequence infinite?

%H Rémy Sigrist, <a href="/A375815/b375815.txt">Table of n, a(n) for n = 1..10000</a>

%e The first terms, alongside the corresponding sums, are:

%e   n   a(n)  Sum {k=1..n} 1/(a(k)*a(n+1-k))

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

%e    1     1  1

%e    2     2  1

%e    3     3  11/12

%e    4     3  1

%e    5     4  17/18

%e    6     4  35/36

%e    7     5  167/180

%e    8     5  14/15

%e    9     5  77/80

%e   10     5  119/120

%e   11     6  77/80

%e   12     6  29/30

%e   13     6  443/450

%e   14     7  3007/3150

%e   15     7  6011/6300

%o (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 Cf. A375814, A375834.

%K nonn

%O 1,2

%A _Rémy Sigrist_, Aug 30 2024