A292961 Rectangular array by antidiagonals: T(n,m) = rank of n*(r+m) when all the numbers k*(r+h), where r = -1+(1+sqrt(5))/2, k>=1, h>=0, are jointly ranked.

1, 3, 2, 6, 8, 4, 9, 15, 13, 5, 12, 22, 25, 19, 7, 17, 30, 38, 35, 27, 10, 20, 40, 52, 54, 48, 33, 11, 24, 49, 66, 74, 72, 61, 41, 14, 28, 58, 82, 93, 98, 91, 73, 46, 16, 32, 67, 96, 115, 124, 122, 108, 85, 55, 18, 37, 78, 111, 136, 151, 155, 146, 129, 101

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    3    6    9    12   17   20
2    8    15   22   30   40   49
4    13   25   38   52   66   82
5    19   35   54   74   93   115
7    27   48   72   98   124  151
10   33   61   91   122  155  190
11   41   73   108  146  187  226
14   46   85   129  172  218  266
The numbers k*(r+h), approximately:
(for k=1):   0.618   1.618   2.618 ...
(for k=2):   1.236   3.236   5.236 ...
(for k=3):   1.854   4.854   7.854 ...
Replacing each k*(r+h) by its rank gives
1    3    6
2    8    15
4    13   25
		

Links

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

Crossrefs

Cf. A182801, A292959, A292960.

Programs

Mathematica

r = -1+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]  (* A292961 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292961 sequence *)

Formula

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