A292958 Rectangular array by antidiagonals: T(n,m) = rank of n*(r+m) when all the numbers k*(r+h), where r = sqrt(5), k>=1, h>=0, are jointly ranked.

1, 2, 4, 3, 7, 8, 5, 11, 14, 12, 6, 16, 21, 22, 17, 9, 20, 29, 33, 30, 24, 10, 26, 38, 44, 45, 40, 28, 13, 32, 47, 57, 61, 59, 51, 35, 15, 37, 56, 69, 77, 80, 73, 60, 41, 18, 43, 66, 84, 94, 101, 97, 88, 71, 49, 19, 50, 76, 99, 113, 123, 124, 115, 103, 82

Offset: 1

Author

Clark Kimberling, Oct 05 2017

Keywords

• nonn
• easy
• tabl

Comments

This is the transpose of the array at A182848. Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers.

Examples

			Northwest corner:
1     2      3      5      6      9     10     13     15
4     7      11     16     20     26    32     37     43
8     14     21     29     38     47    56     66     76
12    22     33     44     57     69    84     99     112
17    30     45     61     77     94    113    132    152
24    40     59     80     101    123   146    169    194
28    51     73     97     124    150   178    206    236
35    60     88     115    147    180   212    247    282
The numbers k*(r+h), approximately:
(for k=1):   2.236   3.236   4.236 ...
(for k=2):   4.472   6.472   6.472 ...
(for k=3):   6.708   9.708   12.708 ...
Replacing each by its rank gives
1      2      3
4      7      11
8      14     21
		

Links

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

Crossrefs

Cf. A182801.

Programs

Mathematica

r = Sqrt[5]; 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]  (* A292958 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292958 sequence *)

Formula

T(n,m) = Sum_{k=1...[n + m*n/r]} [1 - r + n*(r + m)/k], where r=sqrt(5) and [ ]=floor.