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

1,2

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.

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=GoldenRatio and [ ]=floor.

Example

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

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 *)

Crossrefs

Cf. A182801, A292960, A292961.

Sequence in context: A106454 A270195 A297441 * A292957 A338644 A132075

Adjacent sequences: A292956 A292957 A292958 * A292960 A292961 A292962

Keyword

nonn,easy,tabl

Author

Clark Kimberling, Oct 05 2017

Status

approved