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A292958
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Rectangular array by antidiagonals: T(n,m) = rank of n*(r+m) when all the numbers k*(r+h), where r = sqrt(5), k>=1, h>=0, are jointly ranked.
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1
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1, 2, 4, 3, 7, 8, 5, 11, 14, 12, 6, 16, 21, 22, 17, 9, 20, 29, 33, 30, 24, 10, 26, 38, 44, 45, 40, 28, 13, 32, 47, 57, 61, 59, 51, 35, 15, 37, 56, 69, 77, 80, 73, 60, 41, 18, 43, 66, 84, 94, 101, 97, 88, 71, 49, 19, 50, 76, 99, 113, 123, 124, 115, 103, 82
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OFFSET
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1,2
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COMMENTS
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This is the transpose of the array at A182848. Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers.
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LINKS
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FORMULA
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T(n,m) = Sum_{k=1...[n + m*n/r]} [1 - r + n*(r + m)/k], where r=sqrt(5) and [ ]=floor.
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EXAMPLE
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Northwest corner:
1 2 3 5 6 9 10 13 15
4 7 11 16 20 26 32 37 43
8 14 21 29 38 47 56 66 76
12 22 33 44 57 69 84 99 112
17 30 45 61 77 94 113 132 152
24 40 59 80 101 123 146 169 194
28 51 73 97 124 150 178 206 236
35 60 88 115 147 180 212 247 282
The numbers k*(r+h), approximately:
(for k=1): 2.236 3.236 4.236 ...
(for k=2): 4.472 6.472 6.472 ...
(for k=3): 6.708 9.708 12.708 ...
Replacing each by its rank gives
1 2 3
4 7 11
8 14 21
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MATHEMATICA
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r = Sqrt[5]; z = 12;
t[n_, m_] := Sum[Floor[1 - r + n*(r + m)/k], {k, 1, Floor[n + m*n/r]}];
u = Table[t[n, m], {n, 1, z}, {m, 0, z}]; TableForm[u] (* A292958 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten (* A292958 sequence *)
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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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