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

1,2

Comments

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=1/GoldenRatio and [ ]=floor.

Example

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

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 *)

Crossrefs

Cf. A182801, A292959, A292960.

Sequence in context: A348686 A160855 A120232 * A019444 A195412 A069773

Adjacent sequences: A292958 A292959 A292960 * A292962 A292963 A292964

Keyword

nonn,easy,tabl

Author

Clark Kimberling, Oct 05 2017

Status

approved