%I #29 Nov 12 2020 07:22:02

%S 0,1,1,2,0,2,3,1,1,3,4,2,0,2,4,5,3,1,1,3,5,6,4,2,0,2,4,6,7,5,3,1,1,3,

%T 5,7,8,6,4,2,0,2,4,6,8,9,7,5,3,1,1,3,5,7,9,10,8,6,4,2,0,2,4,6,8,10,11,

%U 11,7,5,3,1,1,3,5,7,11,11,12,10,12,6,4,2,0,2,4,6,12,10,12,13,11,11,13,5,3,1,1,3,5,13,11,11,13,14,12,10,12,14,4,2,0,2,4,14,12,10,12,14

%N Square array T(n,k): concatenate the absolute differences of the digits of n and k (the smaller one padded with leading zeros), read by antidiagonals, n, k >= 0.

%C A digit-wise analog of A049581. Referred to as "box" operation by Eric Angelini.

%C 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.

%C 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.

%H Eric Angelini, <a href="https://cinquantesignes.blogspot.com/2019/12/the-box-operation.html">The box ■ operation</a>, personal blog "Cinquante signes", and post to the SeqFan list, Dec 06 2019.

%e The square array starts as follows:

%e n |k=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ...

%e ---+-------------------------------------------------------------

%e 0 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ...

%e 1 | 1 0 1 2 3 4 5 6 7 8 11 10 11 12 13 14 15 16 17 ...

%e 2 | 2 1 0 1 2 3 4 5 6 7 12 11 10 11 12 13 14 15 16 ...

%e 3 | 3 2 1 0 1 2 3 4 5 6 13 12 11 10 11 12 13 14 15 ...

%e 4 | 4 3 2 1 0 1 2 3 4 5 14 13 12 11 10 11 12 13 14 ...

%e 5 | 5 4 3 2 1 0 1 2 3 4 15 14 13 12 11 10 11 12 13 ...

%e 6 | 6 5 4 3 2 1 0 1 2 3 16 15 14 13 12 11 10 11 12 ...

%e 7 | 7 6 5 4 3 2 1 0 1 2 17 16 15 14 13 12 11 10 11 ...

%e 8 | 8 7 6 5 4 3 2 1 0 1 18 17 16 15 14 13 12 11 10 ...

%e 9 | 9 8 7 6 5 4 3 2 1 0 19 18 17 16 15 14 13 12 11 ...

%e 10 | 10 11 12 13 14 15 16 17 18 19 0 1 2 3 4 5 6 7 8 ...

%e 11 | 11 10 11 12 13 14 15 16 17 18 1 0 1 2 3 4 5 6 7 ...

%e 12 | 12 11 10 11 12 13 14 15 16 17 2 1 0 1 2 3 4 5 6 ...

%e (...)

%e It differs from A049581 only if at least one index is > 9.

%e The table of commutators Comm(n,k) := T(T(T(n,k),n),k) reads as follows:

%e 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...

%e ---+---------------------------------------------------------------

%e 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0...

%e 1 | 0 0 2 2 2 2 2 2 2 2 0 0 2 2 2 2 2 2 2 2 0 0 2...

%e 2 | 0 0 0 2 4 4 4 4 4 4 0 0 0 2 4 4 4 4 4 4 0 0 0...

%e 3 | 0 0 0 0 2 4 6 6 6 6 0 0 0 0 2 4 6 6 6 6 0 0 0...

%e 4 | 0 0 0 0 0 2 4 6 8 8 0 0 0 0 0 2 4 6 8 8 0 0 0...

%e 5 | 0 0 0 0 0 0 2 4 6 8 0 0 0 0 0 0 2 4 6 8 0 0 0...

%e 6 | 0 0 0 0 0 0 0 2 4 6 0 0 0 0 0 0 0 2 4 6 0 0 0...

%e 7 | 0 0 0 0 0 0 0 0 2 4 0 0 0 0 0 0 0 0 2 4 0 0 0...

%e 8 | 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0...

%e 9 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0...

%e 10 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 20 20...

%e 11 | 0 0 2 2 2 2 2 2 2 2 0 0 2 2 2 2 2 2 2 2 20 20 22...

%e 12 | 0 0 0 2 4 4 4 4 4 4 0 0 0 2 4 4 4 4 4 4 20 20 20...

%e (...)

%e 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).

%o (PARI) A330240(a,b)=fromdigits(abs(Vec(digits(min(a,b)),if(a+b,-logint(a=max(a,b),10)-1))-digits(a)))

%Y Cf. A330238 (variant excluding row & column 0), A330237 (lower left triangle), A049581 (T(n,k) = |n-k|).

%K nonn,base,tabl

%O 0,4

%A _M. F. Hasler_, Dec 06 2019