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

1,2

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.

Links

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

Formula

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

Example

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

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 *)

Crossrefs

Cf. A182801.

Sequence in context: A365215 A225252 A235201 * A235493 A105081 A235485

Adjacent sequences: A292955 A292956 A292957 * A292959 A292960 A292961

Keyword

nonn,easy,tabl

Author

Clark Kimberling, Oct 05 2017

Status

approved