A330237 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 >= 1.

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, 11, 7, 5, 3, 1, 1, 3, 5, 7, 11, 10, 12, 6, 4, 2, 0, 2, 4, 6, 12, 10, 11, 11, 13, 5, 3, 1, 1, 3, 5, 13, 11, 11, 12, 10, 12, 14, 4, 2, 0, 2, 4, 14, 12, 10, 12, 13, 11, 11, 13, 15, 3, 1, 1, 3, 15, 13, 11, 11, 13, 14, 12, 10, 12, 14, 16, 2, 0, 2, 16

Offset: 1

Author

M. F. Hasler, Dec 06 2019

Keywords

• nonn
• base
• tabl

Comments

A digit-wise analog of A049581.

The binary operator T: N x N -> N is commutative, therefore this table is symmetric and it does not matter in which direction the antidiagonals are read. It would also be sufficient to specify only the lower half of the square table: see A330238 for this variant. The operator is also defined for either argument equal to 0, which is the neutral element: T(x,0) = 0 for all x. Therefore we omit row & column 0 here, see A330240 for the table including these. Every element is its opposite or inverse, as shown by the zero diagonal T(x,x) = 0.

Examples

			The square array starts as follows:
   n | k=1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...
  ---+-----------------------------------------------------------
   1 |   0  1  2  3  4  5  6  7  8 11 10 11 12 13 14 15 16 17 ...
   2 |   1  0  1  2  3  4  5  6  7 12 11 10 11 12 13 14 15 16 ...
   3 |   2  1  0  1  2  3  4  5  6 13 12 11 10 11 12 13 14 15 ...
   4 |   3  2  1  0  1  2  3  4  5 14 13 12 11 10 11 12 13 14 ...
   5 |   4  3  2  1  0  1  2  3  4 15 14 13 12 11 10 11 12 13 ...
   6 |   5  4  3  2  1  0  1  2  3 16 15 14 13 12 11 10 11 12 ...
   7 |   6  5  4  3  2  1  0  1  2 17 16 15 14 13 12 11 10 11 ...
   8 |   7  6  5  4  3  2  1  0  1 18 17 16 15 14 13 12 11 10 ...
   9 |   8  7  6  5  4  3  2  1  0 19 18 17 16 15 14 13 12 11 ...
  10 |  11 12 13 14 15 16 17 18 19  0  1  2  3  4  5  6  7  8 ...
  11 |  10 11 12 13 14 15 16 17 18  1  0  1  2  3  4  5  6  7 ...
  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 > 10.
		

Links

• Eric Angelini, The box ■ operation, personal blog "Cinquante signes", and post to the SeqFan list, Dec 06 2019.
• Eric Angelini, The box ■ operation, personal blog "Cinquante signes", and post to the SeqFan list, Dec 06 2019. [Cached copy]

Crossrefs

Cf A330240 (variant including row & column 0), A330237 (lower left triangle), A049581 (T(n,k) = |n-k|).

Programs

PARI

T(a,b)=fromdigits(abs(Vec(digits(min(a,b)),-logint(a=max(a,b),10)-1)-digits(a)))