A330240 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.

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, 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, 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

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

Table of n, a(n) for n=0..119.

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

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

Sequence in context: A220417 A049581 A114327 * A330237 A231154 A073450

Adjacent sequences: A330237 A330238 A330239 * A330241 A330242 A330243

Keyword

nonn,base,tabl

Author

M. F. Hasler, Dec 06 2019

Status

approved