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A297441
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Rectangular array R by antidiagonals: row n shows the positive integers whose base-3 digits have down-variation n, for n>=0. See Comments.
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4
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1, 2, 3, 4, 7, 6, 5, 9, 15, 33, 8, 10, 18, 57, 60, 13, 11, 19, 61, 141, 303, 14, 12, 20, 69, 168, 519, 546, 17, 16, 21, 87, 177, 543, 1275, 2733, 26, 22, 24, 96, 180, 547, 1518, 4677, 4920, 40, 23, 30, 99, 181, 555, 1599, 4893, 11481, 24603
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OFFSET
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1,2
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COMMENTS
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Suppose that a number n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). See A297330 for a guide to related sequences and partitions of the natural numbers.
Every positive integer occurs exactly once in the array, so that as a sequence this is a permutation of the positive integers.
Conjecture: each column, after some number of initial terms, satisfies the linear recurrence relation c(n) = c(n-1) + 9*c(n-2) - 9*c(n-3).
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LINKS
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EXAMPLE
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Northwest corner:
1 2 4 5 8 13 14 17
3 7 9 10 11 12 16 22
6 15 18 19 20 21 24 30
33 57 61 69 87 96 99 100
60 141 168 177 180 181 182 183
303 519 543 547 555 627 789 870
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MATHEMATICA
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g[n_, b_] := Differences[IntegerDigits[n, b]];
b = 3; z = 200000; u = Table[-Total[Select[g[n, b], # < 0 &]], {n, 1, z}] ;
p[n_] := Position[u, n]; TableForm[Table[Take[Flatten[p[n]], 15], {n, 0, 9}]]
v[n_, k_] := p[k - 1][[n]];
Table[v[k, n - k + 1], {n, 10}, {k, n, 1, -1}] // Flatten
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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