A292960 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)^2, k>=1, h>=0, are jointly ranked.

1, 2, 4, 3, 7, 9, 5, 11, 15, 13, 6, 16, 22, 23, 19, 8, 20, 29, 34, 32, 27, 10, 25, 38, 44, 47, 43, 33, 12, 30, 46, 57, 62, 61, 53, 40, 14, 36, 55, 69, 78, 81, 75, 66, 49, 17, 41, 65, 83, 95, 102, 100, 91, 76, 56, 18, 48, 74, 96, 112, 122, 124, 119, 107, 88

Offset: 1

Author

Clark Kimberling, Oct 05 2017

Keywords

• nonn
• easy
• tabl

Comments

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    8   10
4   7   11   16   20   25  30
9   15  22   29   38   46  55
13  23  34   44   57   69  83
19  32  47   62   78   95  112
27  43  61   81   102  122 145
The numbers k*(r+h), approximately:
(for k=1):   2.618   3.618   4.618 ...
(for k=2):   5.236   7.236   9.236 ...
(for k=3):   7.854   10.854   13.854 ...
Replacing each k*(r+h) by its rank gives
1    2     3
4    7     11
9    15    22
		

Links

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

Crossrefs

Cf. A182801, A292959, A292961.

Programs

Mathematica

r = GoldenRatio^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]  (* A292960 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292960 sequence *)

Formula

T(n,m) = Sum_{k=1...[n + m*n/r]} [1 - r + n*(r + m)/k], where r=(GoldenRatio)^2 and [ ]=floor.