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

1,2

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=(GoldenRatio)^2 and [ ]=floor.

Example

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

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 *)

Crossrefs

Cf. A182801, A292959, A292961.

Sequence in context: A224146 A225010 A235494 * A292963 A262759 A246268

Adjacent sequences: A292957 A292958 A292959 * A292961 A292962 A292963

Keyword

nonn,easy,tabl

Author

Clark Kimberling, Oct 05 2017

Status

approved