A292957 Rectangular array by antidiagonals: T(n,m) = rank of n*(r+m) when all the numbers k*(r+h), where r = sqrt(3), k>=1, h>=0, are jointly ranked.

1, 2, 3, 4, 7, 6, 5, 11, 13, 10, 8, 16, 21, 20, 14, 9, 22, 30, 32, 27, 18, 12, 26, 38, 44, 42, 36, 24, 15, 33, 49, 58, 61, 55, 46, 29, 17, 40, 59, 72, 78, 77, 69, 54, 34, 19, 47, 70, 87, 98, 100, 95, 84, 64, 39, 23, 52, 80, 103, 117, 124, 123, 113, 97, 73

Offset

1,2

Comments

This is the transpose of the array at A182847. 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(3) and [ ]=floor.

Example

Northwest corner:
1    2     4     5     8     9     12    15
3    7     11    16    22    26    33    40
6    13    21    30    38    49    59    70
10   20    32    44    58    72    87    103
14   27    42    61    78    98    117   137
18   36    55    77    100   124   147   175
24   46    69    95    123   152   183   212
The numbers k*(r+h), approximately:
(for k=1):   1.732   2.732    3.732 ...
(for k=2):   3.464   5.464    7.464 ...
(for k=3):   5.196   8.196    12.296 ...
Replacing each by its rank gives
1    2     4
3    7     11
6    13    21

Mathematica

r = Sqrt[3]; 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]  (* A292957 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292957 sequence *)

Crossrefs

Cf. A182801, A182847.

Sequence in context: A270195 A297441 A292959 * A338644 A132075 A265364

Adjacent sequences: A292954 A292955 A292956 * A292958 A292959 A292960

Keyword

nonn,easy,tabl

Author

Clark Kimberling, Oct 05 2017

Status

approved