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A292958
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.
1
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
OFFSET
1,2
COMMENTS
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.
LINKS
FORMULA
T(n,m) = Sum_{k=1...[n + m*n/r]} [1 - r + n*(r + m)/k], where r=sqrt(5) and [ ]=floor.
EXAMPLE
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
MATHEMATICA
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 *)
CROSSREFS
Cf. A182801.
Sequence in context: A365215 A225252 A235201 * A235493 A105081 A235485
KEYWORD
nonn,easy,tabl
AUTHOR
Clark Kimberling, Oct 05 2017
STATUS
approved