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A292961
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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 = -1+(1+sqrt(5))/2, k>=1, h>=0, are jointly ranked.
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3
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1, 3, 2, 6, 8, 4, 9, 15, 13, 5, 12, 22, 25, 19, 7, 17, 30, 38, 35, 27, 10, 20, 40, 52, 54, 48, 33, 11, 24, 49, 66, 74, 72, 61, 41, 14, 28, 58, 82, 93, 98, 91, 73, 46, 16, 32, 67, 96, 115, 124, 122, 108, 85, 55, 18, 37, 78, 111, 136, 151, 155, 146, 129, 101
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OFFSET
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1,2
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COMMENTS
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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=1/GoldenRatio and [ ]=floor.
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EXAMPLE
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Northwest corner:
1 3 6 9 12 17 20
2 8 15 22 30 40 49
4 13 25 38 52 66 82
5 19 35 54 74 93 115
7 27 48 72 98 124 151
10 33 61 91 122 155 190
11 41 73 108 146 187 226
14 46 85 129 172 218 266
The numbers k*(r+h), approximately:
(for k=1): 0.618 1.618 2.618 ...
(for k=2): 1.236 3.236 5.236 ...
(for k=3): 1.854 4.854 7.854 ...
Replacing each k*(r+h) by its rank gives
1 3 6
2 8 15
4 13 25
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MATHEMATICA
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r = -1+GoldenRatio; 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] (* A292961 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten (* A292961 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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