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, 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

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: A247714 A283734 A363162 * A273751 A056017 A091995

Adjacent sequences: A293049 A293050 A293051 * A293053 A293054 A293055

Keyword

nonn,easy,tabl

Author

Clark Kimberling, Oct 06 2017

Status

approved