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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 (list; graph; refs; listen; history; text; internal format)
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 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

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

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Last modified February 27 18:47 EST 2020. Contains 332308 sequences. (Running on oeis4.)