A259899 a(n) is the minimal position at which the maximal value of row n appears in row n of triangle A080936.

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, 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, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13

Offset: 1

Author

Gheorghe Coserea, Jul 07 2015

Keywords

• nonn
• walk

Comments

Empirical: for n>2 there is a unique position at which the maximum of row n occurs.

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).

Examples

			For n=2, a(2)=1 because max{T(2,p), p=1..2}=1 and T(2,1)=1.
For n=4, a(4)=2 because max{T(4,p), p=1..4}=7 and T(4,2)=7.
For n=16, a(16)=5 because max{T(16,p), p=1..16}=9246276 and T(16,5)=9246276.
		

Links

• Gheorghe Coserea, Table of n, a(n) for n = 1..1535

Crossrefs

Cf. A080936, A259885 (value of maximum), A265179.

Formula

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.

a(n) ~ K * sqrt(n), where K = 1.63706... (see A265179). - Gheorghe Coserea, Dec 05 2015