OFFSET
0,2
COMMENTS
The width of the symmetric representation of sigma for hexagonal numbers at the diagonal is 1 only for number 1. For any hexagonal number h(n) = n*(2*n-1), n>1, the last leg of the Dyck path of h(n)-1 has length 2 and that of h(n) has length 1 (see formula in A237591) so that the width of the symmetric representation of sigma at the diagonal is at least 2 and contains a subpart of size 1 at the diagonal (see A280851).
The geometry of the Dyck paths ensures that a square bisected by the diagonal whose side length equals the width of the symmetric representation of sigma at the diagonal fits between the bounding pair of Dyck paths.
For hexagonal numbers up to h(100000) = 19999900000 only 1225, 1413721, and 1631432881 (the 25th, 841st, and 28561st hexagonal numbers) have width 3 at the diagonal, and none were found of odd width greater than 3.
The next [last] number in the sequence data smaller than h(55000) = 6049945000 is a(42) = 4874349480 [a(49) = 4819214400] with a(31..41) > h(55000).
The numbers [1, 1225, 1413721, 1631432881] mentioned above (in the first comment and in the third comment) are the first four square-hexagonal numbers (A046177). - Omar E. Pol, Feb 04 2022
EXAMPLE
a(1) = 6, and a(2) = 120 since all hexagonal numbers k, 6 <= k < 120, have width 2 at the diagonal.
MATHEMATICA
(* parameter bw is an upper bound estimate for how many values will be returned *)
a350712[n_, bw_] := Module[{widthL=Table[0, bw], wL, cL, i, w}, wL=Map[#(2#-1)&, Range[n]]; cL=Map[Last[a2[#]]&, wL]; For[i=1, i<=n, i++, w=cL[[i]]; If[EvenQ[w]&&widthL[[w/2]]==0, widthL[[w/2]]=wL[[i]]]]; Join[{0}, widthL]]
Take[a350712[55000, 50], 37]
CROSSREFS
KEYWORD
nonn
AUTHOR
Hartmut F. W. Hoft, Feb 02 2022
STATUS
approved