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A293052
Rectangular array by antidiagonals: T(n,m) = rank of n*sqrt(3)+m when all the numbers k*sqrt(3)+h, for k >= 1, h >= 0, are jointly ranked.
1
1, 2, 3, 4, 5, 7, 6, 8, 10, 13, 9, 11, 14, 17, 20, 12, 15, 18, 22, 25, 29, 16, 19, 23, 27, 31, 35, 40, 21, 24, 28, 33, 37, 42, 47, 53, 26, 30, 34, 39, 44, 49, 55, 61, 67, 32, 36, 41, 46, 51, 57, 63, 70, 76, 83, 38, 43, 48, 54, 59, 65, 72, 79, 86, 93, 101, 45
OFFSET
1,2
COMMENTS
Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers. As an array, this is the interspersion of sqrt(1/3); see A283962.
LINKS
FORMULA
T(n,m) = Sum_{k=1...n + [m/r]} m+1+[(n-k)r], where r = sqrt(3), [ ]=floor.
EXAMPLE
Northwest corner:
1 2 4 6 9 12 16
3 5 8 11 15 19 24
7 10 14 18 23 28 34
13 17 22 27 33 39 46
20 25 31 37 44 51 59
29 35 42 49 57 65 74
40 47 55 63 72 81 91
53 61 70 79 89 99 110
67 76 86 96 107 118 130
The numbers k*r+h, approximately:
(for k=1): 1.732 2.732 3.732 ...
(for k=2): 3.464 4.464 5.464 ...
(for k=3): 5.196 6.196 7.196 ...
Replacing each k*r+h by its rank gives
1 2 4
3 5 8
7 10 14
MATHEMATICA
r = Sqrt[3]; z = 12;
t[n_, m_] := Sum[Floor[1 + m + (n - k) r], {k, 1, n + Floor[m/r]}];
u = Table[t[n, m], {n, 1, z}, {m, 0, z}]
Grid[u] (* A293052 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten (* A293052 sequence *)
CROSSREFS
Cf. A283962.
Sequence in context: A247714 A283734 A363162 * A273751 A056017 A091995
KEYWORD
nonn,easy,tabl
AUTHOR
Clark Kimberling, Oct 06 2017
STATUS
approved