OFFSET
1,1
COMMENTS
The first and last numbers in row n>=2 are n and 2, respectively, and they occur just once each in the row.
For row n>=3, and if and only if n-1 is prime, numbers n and 2 are the only numbers which occur just once (since when n-1 is prime it cannot make a rectangle for any other number to move from the initial column to the final row).
A number can move twice in succession, and so occur here twice in succession, when it fills the top right corner cell in a rectangle of width * height = n.
The move is from the initial column to top right corner cell, and therefore the numbers which appear twice in succession are d+1 for each divisor d of n, in the range 1 < d < n.
If n is a prime, then it has no such divisors, or if n is a semiprime n = x*y (including square of a prime) then x+1 and y+1 are the only numbers appearing twice in succession.
The length of row n is A002541(n). This equals to the number of special integer partitions of n there. Where a rectangle is formed of the changing shape, the row length increases more because the movement of a number that completes the rectangle is repeated as it continues to move again.
EXAMPLE
The irregular triangle T(n,k) begins:
n/k | 1 2 3 4 5 6 7 8 9 10 11 12 13 14
------------------------------------------------
1 | (empty row)
2 | 2;
3 | 3, 2;
4 | 4, 3, 3, 2;
5 | 5, 4, 3, 4, 2;
6 | 6, 5, 4, 4, 3, 3, 5, 2;
7 | 7, 6, 5, 4, 5, 3, 5, 6, 2;
8 | 8, 7, 6, 5, 5, 4, 6, 3, 3, 4, 7, 2;
9 | 9, 8, 7, 6, 5, 6, 4, 4, 7, 3, 7, 6, 8, 2;
.
Movements of the six-number-high column. 1 never moves. 4 and 3 move twice each in immediate succession as 6 is a composite and a semiprime:
.
6
5 5
4 4 4
3 3 3 3 4 3
2 2 2 5 2 5 2 5 2 5 3 2 5 2
1 1 6 1 6 1 6 1 6 4 1 6 4 1 6 4 3 1 6 4 3 5 1 6 4 3 5 2
.
The parallel is shown for row length and the special integer partition in A002541:
For n = 4, its row consists of 4, 3, 3 and 2, that is four elements.
The special partition of n = 4 is (4), (2 2), (3 1), and (2 1 1), that is also four partitions. The relation is demonstrated by the illustration below. Square blocks represent the four numbers. As they move, the changing shape assumes a number of identical or reflected formations. The number of possible grouping of the blocks within them is exactly the same as the number of the moves that the blocks undergo:
. _ _
| |__________ 1st move
| | _ _ |
| | | |_|____________ 2nd move ____________________________ 4th move
| | | | | _ _ _v_ _|_ |
| | | | | | | |____|___|_____ 3rd move |
| | | |_v_ | | | | |_ _ _v_ _ _ _ _ _ _ _v_
| | | | | | | | | | | | | |
|_ _| |_ _|_ _| |_ _|_ _| |_ _|_ _|_ _| |_ _ _ _ _ _ _ _|
4 3 1 2 2 2 1 1 4
^ ^
|____________________ Identical partition ________________|
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Tamas Sandor Nagy, Mar 19 2023
STATUS
approved