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A292959
Rectangular array by antidiagonals: T(n,m) = rank of n*(r+m) when all the numbers k*(r+h), where r = (1+sqrt(5))/2 (the golden ratio), k>=1, h>=0, are jointly ranked.
3
1, 2, 3, 4, 7, 6, 5, 11, 13, 9, 8, 16, 21, 19, 14, 10, 22, 30, 31, 27, 18, 12, 28, 39, 45, 43, 36, 23, 15, 34, 50, 57, 61, 56, 44, 26, 17, 40, 60, 73, 79, 78, 68, 52, 32, 20, 47, 70, 87, 98, 101, 94, 83, 63, 37, 24, 54, 82, 104, 118, 126, 124, 113, 96, 72
OFFSET
1,2
COMMENTS
This is the transpose of the array at A182849. Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers.
LINKS
FORMULA
T(n,m) = Sum_{k=1...[n + m*n/r]} [1 - r + n*(r + m)/k], where r=GoldenRatio and [ ]=floor.
EXAMPLE
Northwest corner:
1 2 4 5 8 10 12 15
3 7 11 16 22 28 34 40
6 13 21 30 39 50 60 70
9 19 31 45 57 73 87 104
14 27 43 61 79 98 118 138
18 36 56 78 101 126 150 176
23 44 68 94 124 152 184 215
26 52 83 113 146 181 217 255
The numbers k*(r+h), approximately:
(for k=1): 1.618 2.618 3.618 ...
(for k=2): 3.236 5.236 7.236 ...
(for k=3): 4.854 7.854 10.854 ...
Replacing each by its rank gives
1 2 4
3 7 11
6 13 21
MATHEMATICA
r = 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] (* A292959 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten (* A292959 sequence *)
CROSSREFS
KEYWORD
nonn,easy,tabl
AUTHOR
Clark Kimberling, Oct 05 2017
STATUS
approved