%I #14 Dec 07 2019 01:40:04
%S 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,
%T 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,
%U 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
%N 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.
%C A digit-wise analog of A049581.
%C 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.
%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 triangle starts as follows:
%e n | k=1 2 3 4 5 6 7 8 9 10 11
%e ---+-------------------------------------------
%e 1 | 0,
%e 2 | 1, 0,
%e 3 | 2, 1, 0,
%e 4 | 3, 2, 1, 0,
%e 5 | 4, 3, 2, 1, 0,
%e 6 | 5, 4, 3, 2, 1, 0,
%e 7 | 6, 5, 4, 3, 2, 1, 0,
%e 8 | 7, 6, 5, 4, 3, 2, 1, 0,
%e 9 | 8, 7, 6, 5, 4, 3, 2, 1, 0,
%e 10 | 11, 12, 13, 14, 15, 16, 17, 18, 19, 0,
%e 11 | 10, 11, 12, 13, 14, 15, 16, 17, 18, 1, 0,
%e 12 | 11, 10, 11, 12, 13, 14, 15, 16, 17, 2, 1, 0,
%e (...)
%o (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
%Y Cf. A330237 (same as a square array read by antidiagonals), A330240 (idem, including row & column 0), A049581 (T(n,k) = |n-k|).
%K nonn,base
%O 1,4
%A _M. F. Hasler_, Dec 06 2019