A319139 - OEIS

%I #17 Sep 28 2018 23:25:54

%S 4,50,82,1246,581,2494,1744,19961,6981,61136,19210,179669,34935,

%T 122268,57628,244539,96062,2415480,192141,978161,249769,1956341,

%U 576404,2200863,499557

%N 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.

%C 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.

%C 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.

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

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

%e 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).

%e 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).

%t a240542[n_] := Sum[(-1)^(k+1)Ceiling[(n+1)/k-(k+1)/2], {k, 1, Floor[(Sqrt[8n+1]-1)/2]}]

%t (* 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 *)

%t 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]

%t a319139[1,3500000,{},40] (* data *)

%Y Cf. A128605, A237593, A240542, A279286.

%K nonn,more

%O 1,1

%A _Hartmut F. W. Hoft_, Sep 11 2018