A330238 Triangle T(n,k): concatenate the absolute differences of the digits of n and k (the smaller one padded with leading zeros); n >= k >= 1.

0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 3, 2, 1, 0, 5, 4, 3, 2, 1, 0, 6, 5, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 10, 11, 12, 13, 14, 15, 16, 17, 18, 1, 0, 11, 10, 11, 12, 13, 14, 15, 16, 17, 2, 1, 0, 12, 11, 10, 11, 12, 13, 14, 15, 16, 3, 2, 1, 0, 13, 12, 11, 10, 11, 12, 13, 14, 15, 4, 3, 2, 1, 0

Offset: 1

Author

M. F. Hasler, Dec 06 2019

Keywords

• nonn
• base

Comments

A digit-wise analog of A049581.

The binary operator T: N x N -> N is commutative, so we need only the lower half of the symmetric square table A330238 or A330240 (including n, k = 0). Also, 0 is the neutral element: T(x,0) = x for all x, therefore we omit row & column 0. The trivial diagonal T(x,x) = 0 could also be omitted but serves as an end-of-row marker and makes indexing simpler and more natural.

Examples

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

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. A330237 (same as a square array read by antidiagonals), A330240 (idem, including row & column 0), A049581 (T(n,k) = |n-k|).

Programs

PARI

A330238(n,k)=fromdigits(digits(n)-abs(Vec(digits(k),-logint(n,10)-1))) \\ see A330240 for a more general function not limited to 1 <= k <= n