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A330238
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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.
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3
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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
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OFFSET
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1,4
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COMMENTS
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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.
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LINKS
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Eric Angelini, The box ■ operation, personal blog "Cinquante signes", and post to the SeqFan list, Dec 06 2019.
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EXAMPLE
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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,
(...)
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PROG
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(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
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CROSSREFS
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Cf. A330237 (same as a square array read by antidiagonals), A330240 (idem, including row & column 0), A049581 (T(n,k) = |n-k|).
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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