A292959 Rectangular array by antidiagonals: T(n,m) = rank of n*(r+m) when all the numbers k*(r+h), where r = (1+sqrt(5))/2 (the golden ratio), k>=1, h>=0, are jointly ranked.

1, 2, 3, 4, 7, 6, 5, 11, 13, 9, 8, 16, 21, 19, 14, 10, 22, 30, 31, 27, 18, 12, 28, 39, 45, 43, 36, 23, 15, 34, 50, 57, 61, 56, 44, 26, 17, 40, 60, 73, 79, 78, 68, 52, 32, 20, 47, 70, 87, 98, 101, 94, 83, 63, 37, 24, 54, 82, 104, 118, 126, 124, 113, 96, 72

Offset: 1

Author

Clark Kimberling, Oct 05 2017

Keywords

• nonn
• easy
• tabl

Comments

This is the transpose of the array at A182849. 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     10    12    15
3    7    11   16    22    28    34    40
6    13   21   30    39    50    60    70
9    19   31   45    57    73    87    104
14   27   43   61    79    98    118   138
18   36   56   78    101   126   150   176
23   44   68   94    124   152   184   215
26   52   83   113   146   181   217   255
The numbers k*(r+h), approximately:
(for k=1):   1.618   2.618   3.618 ...
(for k=2):   3.236   5.236   7.236 ...
(for k=3):   4.854   7.854   10.854 ...
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, A292960, A292961.

Programs

Mathematica

r = GoldenRatio; 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]  (* A292959 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292959 sequence *)

Formula

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