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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 (list; table; graph; refs; listen; history; text; internal format)
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

Clark Kimberling, Antidiagonals n=1..60, flattened

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: A087465 A247714 A283734 * A273751 A056017 A091995

Adjacent sequences:  A293049 A293050 A293051 * A293053 A293054 A293055

KEYWORD

nonn,easy,tabl

AUTHOR

Clark Kimberling, Oct 06 2017

STATUS

approved

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Last modified June 22 07:09 EDT 2021. Contains 345374 sequences. (Running on oeis4.)