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A297551
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Rectangular array R by antidiagonals: row n shows the positive integers whose base-4 digits have total variation n, for n>=0. See Comments.
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3
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1, 2, 4, 3, 6, 7, 5, 9, 8, 12, 10, 11, 13, 18, 19, 15, 14, 17, 24, 29, 28, 21, 16, 23, 30, 34, 35, 51, 42, 20, 25, 33, 44, 50, 77, 76, 63, 22, 27, 39, 49, 72, 99, 114, 115, 85, 26, 31, 45, 55, 78, 113, 141, 140, 204, 170, 37, 32, 48, 67, 92, 119, 179, 205
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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 a homogeneous linear recurrence relation.
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LINKS
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EXAMPLE
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Northwest corner:
1 2 3 5 10 15 21 42
4 6 9 11 14 16 20 22
7 8 13 17 23 25 27 31
12 18 24 30 33 39 45 48
19 29 34 44 49 55 67 71
28 35 50 72 78 92 98 108
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MATHEMATICA
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a[n_, b_] := Differences[IntegerDigits[n, b]];
b = 4; z = 250000; t = Table[a[n, b], {n, 1, z}];
u = Map[Total, Map[Abs, t]]; 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, 12}, {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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