A292956 Rectangular array by antidiagonals: T(n,m) = rank of n*(r+m) when all the numbers k*(r+h), where r = sqrt(2), k>=1, h>=0, are jointly ranked.

1, 2, 3, 4, 7, 5, 6, 11, 13, 9, 8, 17, 21, 19, 12, 10, 23, 30, 32, 26, 16, 14, 29, 39, 46, 44, 35, 20, 15, 36, 50, 59, 61, 55, 42, 24, 18, 41, 62, 75, 81, 77, 67, 51, 28, 22, 49, 72, 90, 100, 102, 95, 82, 60, 33, 25, 56, 84, 106, 120, 128, 125, 113, 93, 69

Offset: 1

Author

Clark Kimberling, Oct 04 2017

Keywords

• nonn
• easy
• tabl

Comments

This is the transpose of the array at A182846. Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers. [Sequence reference corrected by Peter Munn, Aug 27 2022]

Examples

			Northwest corner:
1   2    4    6    8    10   14   15   18
3   7    11   17   23   29   36   41   49
5   13   21   30   39   50   62   72   84
9   19   32   46   59   75   90   106  124
12  26   44   61   81   100  120  142  165
The numbers k*(r+h), approximately:
(for k=1):   1.414   2.414   3.414 ...
(for k=2):   2.828   4.828   6.828 ...
(for k=3):   4.242   7.242   10.242 ...
Replacing each by its rank gives
1   2    4
3   7    11
5   13   21
		

Links

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

Crossrefs

Cf. A182846.

Programs

Mathematica

r = Sqrt[2]; 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]  (* A292956 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292956 sequence *)

Formula

T(n,m) = Sum_{k=1...[n + m*n/r]} [1 - r + n*(r + m)/k], where r=sqrt(2) and [ ]=floor.