OFFSET
1,2
COMMENTS
The array is doubly periodic (see first formula) and consists of the following repeating 3 X 3 pattern with two components of 1 1's, two components of 2 2's and one component of 3 3's:
+---+-------+
| 1 | 2 2 |
+---+---+---+
| 2 | 1 | 3 |
| +---+ |
| 2 | 3 3 |
+---+-------+
FORMULA
A(n+3, k) = A(n, k+3) = A(n, k).
A(n, k) = A(k, n).
EXAMPLE
Array A(n, k) begins:
n\k | 1 2 3 4 5 6 7 8 9 10
----+-----------------------------
1 | 1 2 2 1 2 2 1 2 2 1
2 | 2 1 3 2 1 3 2 1 3 2
3 | 2 3 3 2 3 3 2 3 3 2
4 | 1 2 2 1 2 2 1 2 2 1
5 | 2 1 3 2 1 3 2 1 3 2
6 | 2 3 3 2 3 3 2 3 3 2
7 | 1 2 2 1 2 2 1 2 2 1
8 | 2 1 3 2 1 3 2 1 3 2
9 | 2 3 3 2 3 3 2 3 3 2
10 | 1 2 2 1 2 2 1 2 2 1
.
We can chose A(1, 1) = 1.
A(2, 1) cannot equal 1; we chose A(2, 1) = 2.
Likewise we chose A(1, 2).
A(2, 2) cannot equal 2 as this would imply a component with 3 or more 2's.
So, by necessity, we chose A(3, 1) = A(1, 3) = 2.
We chose A(2, 2) = 1.
We chose A(4, 1) = 1.
A(3, 2) cannot equal 1 or 2; we chose A(3, 2) = 3.
Likewise we chose A(2, 3) = 3.
We chose A(1, 4) = 1.
A(5, 1) cannot equal 1; we chose A(5, 1) = 2.
A(4, 2) cannot equal 1 (or 3); we chose A(4, 2) = 2.
By necessity, A(3, 3) = 3.
etc.
PROG
(PARI) A(n, k) = { [1, 2, 2; 2, 1, 3; 2, 3, 3][1+(n-1)%3, 1+(k-1)%3] }
CROSSREFS
KEYWORD
AUTHOR
Rémy Sigrist, Dec 13 2023
STATUS
approved