OFFSET
1,2
COMMENTS
FORMULA
a((w+p-2)(w+p-1)/2 + p) = T(w,p), for all w, p >= 1.
T(w(n),p(n)) = a(n), for all n >= 1, where p(n) = n - r(n-1) * (r(n-1) + 1)/2), w(n) = r(n-1) - p(n) + 2, and r(n) = floor((sqrt(8*n+1) - 1)/2).
EXAMPLE
The 10x8 section of the table T(w,p) with dashes indicating values greater than 120*10^6; rows w denote the common maximum width in all parts and columns p the number of parts in the symmetric representation of sigma(T(w,p)).
w\p | 1 2 3 4 5 6 7 8 ...
--------------------------------------------------------------------------
1 | 1 3 9 21 81 147 729 903
2 | 6 78 1014 12246 171366 1922622 28960854 -
3 | 60 7620 967740 116136420 - - -
4 | 120 28920 6969720 -
5 | 360 261720 -
6 | 840 1422120 -
7 | 3360 22622880 -
8 | 2520 12728520 -
9 | 5040 50858640 -
10| 10080 -
...
The symmetric representation of sigma for T(2,2) = 78 consists of the two parts (84, 84) of maximum widths (2, 2), and that of T(2,3) = 1014 consists of the three parts (1020, 156, 1020) of maximum widths (2, 2, 2).
MATHEMATICA
(* function a341969 is defined in A341969 *)
a348142[n_, {w_, p_}] := Module[{list=Table[0, {i, w}, {j, p}], k, s, c, u}, For[k=1, k<=n, k++, s=Map[Max, Select[SplitBy[a341969[k], #!=0&], #[[1]]!=0&]]; c=Length[s]; u=Union[s]; If[Length[u]==1&&u[[1]]<=w&&c<=p, If[list[[u[[1]], c]]==0, list[[u[[1]], c]]=k]]]; list]
table=a348142[12000000}, {10, 10}] (* 10x10 table; very long computation time *)
p[n_] := n-row[n-1](row[n-1]+1)/2
w[n_] := row[n-1]-p[n]+2
Map[table[[w[#], p[#]]]&, Range[23]] (* sequence data *)
CROSSREFS
KEYWORD
AUTHOR
Hartmut F. W. Hoft, Oct 04 2021
STATUS
approved