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%I #44 Dec 21 2015 01:52:36
%S 1,1,2,2,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,
%T 8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,
%U 11,12,12,12,12,12,12,12,12,12,12,13,13,13,13
%N a(n) is the minimal position at which the maximal value of row n appears in row n of triangle A080936.
%C Empirical: for n>2 there is a unique position at which the maximum of row n occurs.
%C Conjecture: a(n) = floor(sqrt(p*n+q)+r) for all n>=1, where p = 2.67996... = A265179^2 and q,r are some constants (best values found: q=3.6, r=-1).
%H Gheorghe Coserea, <a href="/A259899/b259899.txt">Table of n, a(n) for n = 1..1535</a>
%F a(n) = min argmax(k->T(n,k), k=1..n), that is a(n) = min{k, T(n,k) = max{T(n,p), p=1..n}}, where T(n,k) is the number of Dyck paths of length 2n and height k, 1 <= k <= n.
%F a(n) ~ K * sqrt(n), where K = 1.63706... (see A265179). - _Gheorghe Coserea_, Dec 05 2015
%e For n=2, a(2)=1 because max{T(2,p), p=1..2}=1 and T(2,1)=1.
%e For n=4, a(4)=2 because max{T(4,p), p=1..4}=7 and T(4,2)=7.
%e For n=16, a(16)=5 because max{T(16,p), p=1..16}=9246276 and T(16,5)=9246276.
%Y Cf. A080936, A259885 (value of maximum), A265179.
%K nonn,walk
%O 1,3
%A _Gheorghe Coserea_, Jul 07 2015
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