

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.


4



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
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OFFSET

1,2


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 verticallinesegment 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


LINKS

Table of n, a(n) for n=1..35.


EXAMPLE

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.


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) *)


CROSSREFS

Cf. A128605, A237593, A240542, A245092, A279286.
Sequence in context: A200862 A096690 A279286 * A050612 A120110 A047694
Adjacent sequences: A282194 A282195 A282196 * A282198 A282199 A282200


KEYWORD

nonn,hard,more


AUTHOR

Hartmut F. W. Hoft, Feb 08 2017


STATUS

approved



