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A350712 a(n) is the smallest hexagonal number for which the symmetric representation of sigma(n) has width 2*n, n >= 0, at the diagonal.

%I A350712 #41 Mar 05 2022 00:35:09
%S A350712 0,6,120,2016,7140,61776,103740,738720,437580,1185030,4680270,4426800,
%T A350712 2031120,6193440,4915680,30728880,2162160,48565440,134734320,
%U A350712 286071240,163723560,376902240,536592420,137373600,76576500,391986000,214980480,103672800,1018606680,5401294080
%N A350712 a(n) is the smallest hexagonal number for which the symmetric representation of sigma(n) has width 2*n, n >= 0, at the diagonal.
%C A350712 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).
%C A350712 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.
%C A350712 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.
%C A350712 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).
%C A350712 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
%e A350712 a(1) = 6, and a(2) = 120 since all hexagonal numbers k, 6 <= k < 120, have width 2 at the diagonal.
%t A350712 (* for function a2[ ] see A237048 and A249223 *)
%t A350712 (* parameter bw is an upper bound estimate for how many values will be returned *)
%t A350712 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]]
%t A350712 Take[a350712[55000, 50], 37]
%Y A350712 Cf. A000384, A046177, A235791, A237048, A237591, A237593, A249223, A280851, A346875.
%K A350712 nonn
%O A350712 0,2
%A A350712 _Hartmut F. W. Hoft_, Feb 02 2022