A293056 Rectangular array by antidiagonals: T(n,m) = rank of n*log(2)+m when all the numbers k*log(2)+h, for k >= 1, h >= 0, are jointly ranked.

1, 3, 2, 6, 5, 4, 11, 9, 8, 7, 17, 15, 13, 12, 10, 25, 22, 20, 18, 16, 14, 34, 31, 28, 26, 23, 21, 19, 45, 41, 38, 35, 32, 29, 27, 24, 57, 53, 49, 46, 42, 39, 36, 33, 30, 70, 66, 62, 58, 54, 50, 47, 43, 40, 37, 85, 80, 76, 72, 67, 63, 59, 55, 51, 48, 44, 101

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 1/log(2); see A283962.

Examples

			Northwest corner:
1      3      6      11     17     25     34
2      5      9      15     22     31     41
4      8      13     20     28     38     49
7      12     18     26     35     46     58
10     16     23     32     42     54     67
14     21     29     39     50     63     77
19     27     36     47     59     73     88
24     33     43     55     68     83     99
30     40     51     64     78     94     111
The numbers k*r+h, approximately:
(for k=1):   0.693   1.693   2.693 ...
(for k=2):   1.386   2.386   3.386 ...
(for k=3):   2.079   3.079   4.079 ...
Replacing each k*r+h by its rank gives
1    3    6
2    5    9
4    8    13
		

Links

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

Crossrefs

Cf. A283962.

Programs

Mathematica

r = Log[2]; 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] (* A293056 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A293056 sequence *)

Formula

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