A293054 Rectangular array by antidiagonals: T(n,m) = rank of n*sqrt(5)+m when all the numbers k*sqrt(5)+h, for k >= 1, h >= 0, are jointly ranked.

1, 2, 4, 3, 6, 9, 5, 8, 12, 16, 7, 11, 15, 20, 25, 10, 14, 19, 24, 30, 37, 13, 18, 23, 29, 35, 43, 51, 17, 22, 28, 34, 41, 49, 58, 67, 21, 27, 33, 40, 47, 56, 65, 75, 85, 26, 32, 39, 46, 54, 63, 73, 83, 94, 106, 31, 38, 45, 53, 61, 71, 81, 92, 103, 116, 129

Offset: 1

Author

Clark Kimberling, Oct 06 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. As an array, this is the interspersion of sqrt(1/5); see A283962.

Examples

			Northwest corner:
1    2    3    5    7    10   13
4    6    8    11   14   18   22
9    12   15   19   23   28   33
16   20   24   29   34   40   46
25   30   35   41   47   54   61
37   43   49   56   63   71   79
51   58   65   73   81   90   99
67   75   83   92   101  111  121
85   94   103  113  123  134  145
The numbers k*r+h, approximately:
(for k=1):   2.236   3.236   3.236 ...
(for k=2):   4.472   5.472   6.472 ...
(for k=3):   6.708   7.708   8.708 ...
Replacing each k*r+h by its rank gives
1   2   3
4   6   8
9   12  15
		

Links

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

Crossrefs

Cf. A283962.

Programs

Mathematica

r = Sqrt[5]; z = 12;
t[n_, m_] := Sum[Floor[1 + m + (n - k) r], {k, 1, n + Floor[m/r]}];
u = Table[t[n, m], {n, 1, z}, {m, 0, z}]
Grid[u] (* A293054 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A293054 sequence *)

Formula

T(n,m) = Sum_{k=1...n + [m/r]} m+1+[(n-k)r], where r = sqrt(5), [ ]=floor.