OFFSET
0,4
COMMENTS
A digit-wise analog of A049581. Referred to as "box" operation by Eric Angelini.
The binary operator T: N x N -> N is commutative, so this table is symmetric: it does not matter in which direction the antidiagonals are read, and it would be sufficient to specify only the lower half of the square table: see A330238 for this triangle. Zero is the neutral element: T(x,0) = x for all x. Any x is its own inverse or opposite x', as shown by the zero diagonal T(x,x) = 0.
A measure of non-associativity is the "commutator" ((x T y) T x') T y' = ((x T y) T x) T y which would be zero in the associative case, given that x = x' for all x. Here it turns out to be given by 2*A053616, read as a triangle, and rows extended quasi-periodically with period 10, see example.
LINKS
Eric Angelini, The box ■ operation, personal blog "Cinquante signes", and post to the SeqFan list, Dec 06 2019.
EXAMPLE
The square array starts as follows:
n |k=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ...
---+-------------------------------------------------------------
0 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ...
1 | 1 0 1 2 3 4 5 6 7 8 11 10 11 12 13 14 15 16 17 ...
2 | 2 1 0 1 2 3 4 5 6 7 12 11 10 11 12 13 14 15 16 ...
3 | 3 2 1 0 1 2 3 4 5 6 13 12 11 10 11 12 13 14 15 ...
4 | 4 3 2 1 0 1 2 3 4 5 14 13 12 11 10 11 12 13 14 ...
5 | 5 4 3 2 1 0 1 2 3 4 15 14 13 12 11 10 11 12 13 ...
6 | 6 5 4 3 2 1 0 1 2 3 16 15 14 13 12 11 10 11 12 ...
7 | 7 6 5 4 3 2 1 0 1 2 17 16 15 14 13 12 11 10 11 ...
8 | 8 7 6 5 4 3 2 1 0 1 18 17 16 15 14 13 12 11 10 ...
9 | 9 8 7 6 5 4 3 2 1 0 19 18 17 16 15 14 13 12 11 ...
10 | 10 11 12 13 14 15 16 17 18 19 0 1 2 3 4 5 6 7 8 ...
11 | 11 10 11 12 13 14 15 16 17 18 1 0 1 2 3 4 5 6 7 ...
12 | 12 11 10 11 12 13 14 15 16 17 2 1 0 1 2 3 4 5 6 ...
(...)
It differs from A049581 only if at least one index is > 9.
The table of commutators Comm(n,k) := T(T(T(n,k),n),k) reads as follows:
n |k=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22...
---+---------------------------------------------------------------
0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0...
1 | 0 0 2 2 2 2 2 2 2 2 0 0 2 2 2 2 2 2 2 2 0 0 2...
2 | 0 0 0 2 4 4 4 4 4 4 0 0 0 2 4 4 4 4 4 4 0 0 0...
3 | 0 0 0 0 2 4 6 6 6 6 0 0 0 0 2 4 6 6 6 6 0 0 0...
4 | 0 0 0 0 0 2 4 6 8 8 0 0 0 0 0 2 4 6 8 8 0 0 0...
5 | 0 0 0 0 0 0 2 4 6 8 0 0 0 0 0 0 2 4 6 8 0 0 0...
6 | 0 0 0 0 0 0 0 2 4 6 0 0 0 0 0 0 0 2 4 6 0 0 0...
7 | 0 0 0 0 0 0 0 0 2 4 0 0 0 0 0 0 0 0 2 4 0 0 0...
8 | 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0...
9 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0...
10 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 20 20...
11 | 0 0 2 2 2 2 2 2 2 2 0 0 2 2 2 2 2 2 2 2 20 20 22...
12 | 0 0 0 2 4 4 4 4 4 4 0 0 0 2 4 4 4 4 4 4 20 20 20...
(...)
Up to row & column 10, the columns are twice the sequence A053616 written as triangle. The first 10 X 10 block repeats horizontally and vertically. Further away from the origin, the elements of this block multiplied by corresponding powers of 10 are added to the corresponding 10 X 10 blocks: e.g., the block Comm(130..139,270..279) = Comm(0..9,0..9) + 260, where 260 = 100*Comm(1,2) + 10*Comm(3,7).
PROG
(PARI) A330240(a, b)=fromdigits(abs(Vec(digits(min(a, b)), if(a+b, -logint(a=max(a, b), 10)-1))-digits(a)))
CROSSREFS
KEYWORD
AUTHOR
M. F. Hasler, Dec 06 2019
STATUS
approved