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

1,4

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.

Links

Table of n, a(n) for n=1..115.

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=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.

Prog

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

Crossrefs

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

Sequence in context: A049581 A114327 A330240 * A231154 A073450 A284592

Adjacent sequences: A330234 A330235 A330236 * A330238 A330239 A330240

Keyword

nonn,base,tabl

Author

M. F. Hasler, Dec 06 2019

Status

approved