OFFSET
1,2
COMMENTS
T(1,k+1) <= 3^k, for all k>=0, since for k=2j the (j+1)-st part in the symmetric representation of sigma(3^k) extends across the diagonal, and for k=2j+1 the (j+1)-st part is completed before the diagonal.
The data computed so far for a partially filled table of 15 rows and 15 columns, show that all rows, all columns (except column 4 for n <= 6 *10^7), and the diagonal are nonmonotonic.
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 10x10 section of the table with dashes indicating values greater than 6*10^7; rows w denote the maximum width 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 9 ...
----------------------------------------------------------------------------
1 | 1 3 9 21 81 147 729 903 3025
2 | 6 78 15 75 63 189 357 1197 2499
3 | 60 7620 315 495 525 1275 1287 3861 3591
4 | 120 28920 1326 22542 5005 16575 2145 29325 11583
5 | 360 261720 3465 44574 6435 131835 76125 24225 82593
6 | 840 1422120 17325 199578 34034 83655 196707 468027 62985
7 | 3360 22622880 45045 7734558 153153 442442 314925 1108965 471975
8 | 2520 12728520 51975 - 205275 2067065 1429275 2359875 557175
9 | 5040 50858640 225225 - 646646 2863718 2395197 5353725 2785875
10| 10080 - 405405 - 1990989 2124694 6500375 36535499 7753875
...
The symmetric representation of sigma for T(2,3)) = 15 consists of the three parts (8, 8, 8) of maximum widths (1, 2, 1), and that of T(3,3) = 315 consists of the three parts (158, 308, 158) of maximum widths (1, 3, 1).
MATHEMATICA
(* function a341969 is defined in A341969 *)
a348171[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 = Max[s]; If[u<=w && c<=p, If[list[[u, c]] == 0, list[[u, c]] = k ]]]; list]
table=a348171[60000000, {15, 15}] (* 15x15 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[55]] (* sequence data *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Hartmut F. W. Hoft, Oct 04 2021
STATUS
approved