A292964 Rectangular array by antidiagonals: T(n,m) = rank of n*(1/e + m) when all the numbers k*(1/e+h), for k>=1, h>=0, are jointly ranked.

1, 4, 2, 8, 10, 3, 13, 19, 16, 5, 17, 29, 32, 23, 6, 22, 40, 48, 44, 30, 7, 27, 52, 65, 68, 58, 37, 9, 34, 63, 82, 93, 89, 72, 46, 11, 38, 76, 102, 118, 120, 108, 87, 53, 12, 43, 88, 123, 144, 153, 149, 132, 101, 60, 14, 50, 99, 141, 171, 187, 189, 178, 155

Offset: 1

Author

Clark Kimberling, Oct 05 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.

Examples

			Northwest corner:
1      4     8      13     17     22
2     10     19     29     40     52
3     16     32     48     65     82
5     23     44     68     93     118
6     30     58     89     120    153
7     37     72     108    149    189
9     46     87     132    178    228
The numbers k*(1/e+h), approximately:
(for k=1):   0.367   1.367  2.3667 ...
(for k=2):   0.735   2.735  4.735 ...
(for k=3):   1.103   4.103  7.103 ...
Replacing each by its rank gives
1    4     8
2    10    19
3    16    32
		

Links

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

Crossrefs

Cf. A182801, A292963.

Programs

Mathematica

r = 1/E; 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]  (* A292964 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292964 sequence *)

Formula

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