cp's OEIS Frontend

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.

%I A319139 #21 Jun 19 2025 09:19:08
%S A319139 4,50,82,1246,581,2494,1744,19961,6981,61136,19210,179669,34935,
%T A319139 122268,57628,244539,96062,2415480,192141,978161,249769,1956341,
%U A319139 576404,2200863,499557
%N 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.
%C A319139 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 A319139 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 A319139 Additional values of this sequence: a(27) = 1152829, a(29) = 999115, a(31) = 1498678, a(33) = 2305659.
%F A319139 It appears that a(n) = A240542(A128605(n+1)-1) + 1.
%e A319139 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 A319139 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 A319139 a240542[n_] := Sum[(-1)^(k+1)Ceiling[(n+1)/k-(k+1)/2], {k, 1, Floor[(Sqrt[8n+1]-1)/2]}]
%t A319139 (* 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 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 a319139[1,3500000,{},40] (* data *)
%Y A319139 Cf. A128605, A237593, A240542, A279286.
%K A319139 nonn,more
%O A319139 1,1
%A A319139 _Hartmut F. W. Hoft_, Sep 11 2018