A292956 Rectangular array by antidiagonals: T(n,m) = rank of n*(r+m) when all the numbers k*(r+h), where r = sqrt(2), k>=1, h>=0, are jointly ranked.

1, 2, 3, 4, 7, 5, 6, 11, 13, 9, 8, 17, 21, 19, 12, 10, 23, 30, 32, 26, 16, 14, 29, 39, 46, 44, 35, 20, 15, 36, 50, 59, 61, 55, 42, 24, 18, 41, 62, 75, 81, 77, 67, 51, 28, 22, 49, 72, 90, 100, 102, 95, 82, 60, 33, 25, 56, 84, 106, 120, 128, 125, 113, 93, 69

Offset

1,2

Comments

This is the transpose of the array at A182846. Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers. [Sequence reference corrected by Peter Munn, Aug 27 2022]

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(2) and [ ]=floor.

Example

Northwest corner:
1   2    4    6    8    10   14   15   18
3   7    11   17   23   29   36   41   49
5   13   21   30   39   50   62   72   84
9   19   32   46   59   75   90   106  124
12  26   44   61   81   100  120  142  165
The numbers k*(r+h), approximately:
(for k=1):   1.414   2.414   3.414 ...
(for k=2):   2.828   4.828   6.828 ...
(for k=3):   4.242   7.242   10.242 ...
Replacing each by its rank gives
1   2    4
3   7    11
5   13   21

Mathematica

r = Sqrt[2]; 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]  (* A292956 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292956 sequence *)

Crossrefs

Cf. A182846.

Sequence in context: A071651 A072658 A099864 * A056535 A026237 A308301

Adjacent sequences: A292953 A292954 A292955 * A292957 A292958 A292959

Keyword

nonn,easy,tabl

Author

Clark Kimberling, Oct 04 2017

Status

approved