# This is the a326017.txt text file.
# Conjectures on columns of T(n,k) = number of knapsack partitions of n with maximum k, from Fausto A. C. Cariboni, Jun 05 2021

Let a_k(n) be the number of knapsack partitions of n with largest part k constant.
By analyzing sequences a_k(n) it was possible to identify the existence of repeated subsequences 
of consecutive terms where both their length and starting point seem to be easy to calculate.

Let SP(k) be the starting point of the first cycle.
Let LC(k) be the length of the cycle (repeated subsequence).
Let LS(k) be the number of computed terms of the sequence a_k(n).

 k   SP(k)   LC(k)   LS(k)     OEIS
 2     2        2
 3     8        6     5000   A326034
 4    18       12    10000   A344310
 5    32       60    10000   A343321
 6    50       60    10000   A344340
 7    72      420    25000   A344412
 8    98      840    40000   A342684
 9   128     2520    50000   A344625
10   162     2520    50000   A344635
11   200    27720    60000
12   242    27720    60000

Conjecture: The starting point of the first cycle is computable with the following rule.
SP(k) = 2, if k = 2.
SP(k) = SP(k-1) + 2 + 4 * (k-2), for k > 2.

Conjecture: The length of the cycle is computable with the following rule.
LC(k) = 2, if k = 2.
LC(k) = p * LC(k-1), if k = p^s, p prime and s > 0.
LC(k) = LC(k-1), otherwise.