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

Author

Clark Kimberling, Oct 06 2017

Keywords

• nonn
• easy
• tabl

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.

Examples

			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
		

Links

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

Crossrefs

Cf. A283962.

Programs

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 *)

Formula

T(n,m) = Sum_{k=1...n + [m/r]} m+1+[(n-k)r], where r = sqrt(3), [ ]=floor.