A282197 a(n) is the smallest number d if the point (d,d) is shared by exactly n different Dyck paths in the main diagonal of the diagram of the symmetries of sigma described in A237593.

1, 2, 7, 15, 52, 102, 296, 371, 455, 929, 1853, 2034, 4517, 4797, 5829, 6146, 6948, 17577, 19818, 18915, 60349, 78369, 113010, 110185, 91650, 85171, 311321, 123788, 823049, 128596, 1650408, 1136865, 415355, 906771, 2897535

Offset: 1

Author

Hartmut F. W. Hoft, Feb 08 2017

Keywords

• nonn
• hard
• more

Comments

This sequence is not monotone since a(19) = 19818 > 18915 = a(20).

Additional values smaller than 5000000 are a(37) = 1751785, a(38) = 1786732, a(39) = 1645139, a(41) = 1308771 and a(44) = 3329668.

Sequence A128605 of first occurrences of gaps between adjacent Dyck paths appears to be unrelated to this sequence.

First differs from A279286 (which is monotone) at a(19). - Omar E. Pol, Feb 08 2017

a(n) = d if the point (d,d) belongs to the first vertical-line-segment of exactly length n found in the main diagonal of the pyramid described in A245092 (starting from the top). The diagram of the symmetries of sigma is also the top view of the pyramid. - Omar E. Pol, Feb 09 2017

Examples

			The four examples listed in A279286 are also examples for this sequences.
a(20) = 18915 is in the sequence since it is the first time that exactly 20 Dyck paths meet on the diagonal though a concurrence of exactly 19 paths on the diagonal happens only later at a(19) = 19818.
		

Crossrefs

Cf. A128605, A237593, A240542, A245092, A279286.

Programs

Mathematica

a240542[n_] := Sum[(-1)^(k+1)*Ceiling[(n+1)/k - (k+1)/2], {k, 1, Floor[(Sqrt[8n+1]-1)/2]}]
(* parameter cL must be sufficiently large for bound b *)
a282197[cL_, b_] := Module[{centers=Map[0&, Range[cL]], acc={1}, k=2, cPrev=1, cCur, len}, While[k<=b, cCur=a240542[k]; If[Last[acc]==cCur, AppendTo[acc,cCur], len=Length[acc]; If[centers[[len]]==0, centers[[len]]=cPrev]; acc={cCur}; cPrev=cCur]; k++]; centers]
a282197[50, 5000000] (* data *)
(* list processing implementation useful for "small" arguments only *)
a282197F[n_] := Map[Last, Sort[Normal[Map[First[First[#]]&, GroupBy[Split[Map[a240542, Range[n]]], Length[#]&]]]]]
a282197F[50000] (* computes a(1) .. a(20) *)