A292957 Rectangular array by antidiagonals: T(n,m) = rank of n*(r+m) when all the numbers k*(r+h), where r = sqrt(3), k>=1, h>=0, are jointly ranked.

1, 2, 3, 4, 7, 6, 5, 11, 13, 10, 8, 16, 21, 20, 14, 9, 22, 30, 32, 27, 18, 12, 26, 38, 44, 42, 36, 24, 15, 33, 49, 58, 61, 55, 46, 29, 17, 40, 59, 72, 78, 77, 69, 54, 34, 19, 47, 70, 87, 98, 100, 95, 84, 64, 39, 23, 52, 80, 103, 117, 124, 123, 113, 97, 73

Offset: 1

Author

Clark Kimberling, Oct 05 2017

Keywords

• nonn
• easy
• tabl

Comments

This is the transpose of the array at A182847. Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers.

Examples

			Northwest corner:
1    2     4     5     8     9     12    15
3    7     11    16    22    26    33    40
6    13    21    30    38    49    59    70
10   20    32    44    58    72    87    103
14   27    42    61    78    98    117   137
18   36    55    77    100   124   147   175
24   46    69    95    123   152   183   212
The numbers k*(r+h), approximately:
(for k=1):   1.732   2.732    3.732 ...
(for k=2):   3.464   5.464    7.464 ...
(for k=3):   5.196   8.196    12.296 ...
Replacing each by its rank gives
1    2     4
3    7     11
6    13    21
		

Links

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

Crossrefs

Cf. A182801, A182847.

Programs

Mathematica

r = Sqrt[3]; 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]  (* A292957 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292957 sequence *)

Formula

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