A293056 - OEIS

%I #6 Oct 06 2017 21:34:46

%S 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,

%T 21,19,45,41,38,35,32,29,27,24,57,53,49,46,42,39,36,33,30,70,66,62,58,

%U 54,50,47,43,40,37,85,80,76,72,67,63,59,55,51,48,44,101

%N 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.

%C 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.

%H Clark Kimberling, <a href="/A293056/b293056.txt">Antidiagonals n=1..60, flattened</a>

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

%e Northwest corner:

%e 1      3      6      11     17     25     34

%e 2      5      9      15     22     31     41

%e 4      8      13     20     28     38     49

%e 7      12     18     26     35     46     58

%e 10     16     23     32     42     54     67

%e 14     21     29     39     50     63     77

%e 19     27     36     47     59     73     88

%e 24     33     43     55     68     83     99

%e 30     40     51     64     78     94     111

%e The numbers k*r+h, approximately:

%e (for k=1):   0.693   1.693   2.693 ...

%e (for k=2):   1.386   2.386   3.386 ...

%e (for k=3):   2.079   3.079   4.079 ...

%e Replacing each k*r+h by its rank gives

%e 1    3    6

%e 2    5    9

%e 4    8    13

%t r = Log[2]; z = 12;

%t t[n_, m_] := Sum[Floor[1 + m + (n - k) r], {k, 1, n + Floor[m/r]}];

%t u = Table[t[n, m], {n, 1, z}, {m, 0, z}]

%t Grid[u] (* A293056 array *)

%t Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A293056 sequence *)

%Y Cf. A283962.

%K nonn,easy,tabl

%O 1,2

%A _Clark Kimberling_, Oct 06 2017