A319139 a(n) is the smallest position k >= 1 on the diagonal at which a record gap of size n between two adjacent Dyck paths of the symmetric representation of sigma starts.

4, 50, 82, 1246, 581, 2494, 1744, 19961, 6981, 61136, 19210, 179669, 34935, 122268, 57628, 244539, 96062, 2415480, 192141, 978161, 249769, 1956341, 576404, 2200863, 499557

Offset

1,1

Comments

This sequence of positions of record gaps on the diagonal is not increasing, in contrast to the apparently increasing sequence A279286 of record numbers of Dyck paths jointly crossing the diagonal.

For n >= 2 it appears that a(2*n) > a(2*n+1), however a(2*n) < a(2*n+2) is false as a(12) = 179669 and a(14) = 122268 show, just as a(2n-1) < a(2*n+1) is false as a(23) = 576404 and a(25) = 499557 show.

Additional values of this sequence: a(27) = 1152829, a(29) = 999115, a(31) = 1498678, a(33) = 2305659.

Links

Table of n, a(n) for n=1..25.

Formula

It appears that a(n) = A240542(A128605(n+1)-1) + 1.

Example

A240542(119) = 81 and A240542(120) = A240542(A128605(4)) = 85 establish the starting position on the diagonal of the first gap of size 3 as 82 = a(3).
A240542(3484799) = 2415479 and A240542(3484800) = A240542(A128605(19)) = 2415498 establish the starting position on the diagonal of the first gap of size 18 as 2415480 = a(18).

Mathematica

a240542[n_] := Sum[(-1)^(k+1)Ceiling[(n+1)/k-(k+1)/2], {k, 1, Floor[(Sqrt[8n+1]-1)/2]}]
(* parameter recs is the list of elements of the sequence in interval 1..m-1 already computed with an entry of 0 representing an element not yet found *)
a319139[m_, n_, recs_, ext_] := Module[{list=Join[recs, Table[0, ext]], a=a240542[m], i, b, g}, For[i=m+1, i<=n, i++, b=a240542[i]; g=b-a-1; If[g>0 && list[[g]==0, list[[g]]=a+1]; a=b]; list]
a319139[1, 3500000, {}, 40] (* data *)

Crossrefs

Cf. A128605, A237593, A240542, A279286.

Sequence in context: A027533 A048559 A356082 * A254543 A321590 A274872

Adjacent sequences: A319136 A319137 A319138 * A319140 A319141 A319142

Keyword

nonn,more

Author

Hartmut F. W. Hoft, Sep 11 2018

Status

approved