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A297553
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Rectangular array R by antidiagonals: row n shows the positive integers whose base-4 digits have up-variation n, for n>=0. See Comments.
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3
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1, 2, 6, 3, 11, 7, 4, 17, 18, 19, 5, 22, 23, 35, 99, 8, 24, 27, 51, 114, 115, 9, 25, 28, 67, 119, 291, 307, 10, 26, 29, 71, 179, 306, 563, 1587, 12, 33, 30, 75, 275, 311, 819, 1827, 1843, 13, 38, 31, 76, 290, 371, 1075, 1842, 4659, 4915
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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 4 5 8 9 10
6 11 17 22 24 25 26 33
7 18 23 27 28 29 30 31
19 35 51 67 71 75 76 77
99 114 119 179 275 290 295 305
115 291 306 311 371 435 451 455
307 563 819 1075 1139 1203 1219 1223
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MATHEMATICA
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g[n_, b_] := Differences[IntegerDigits[n, b]];
b = 4; 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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