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

1,2

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 1/log(2); 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 = log(2) and [ ]=floor.

Example

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

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 *)

Crossrefs

Cf. A283962.

Sequence in context: A277881 A145522 A283939 * A131968 A191740 A132665

Adjacent sequences: A293053 A293054 A293055 * A293057 A293058 A293059

Keyword

nonn,easy,tabl

Author

Clark Kimberling, Oct 06 2017

Status

approved