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
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
KEYWORD
nonn,hard,more
AUTHOR
Hartmut F. W. Hoft, Feb 08 2017
STATUS
approved