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

1,2

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.

Links

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

Formula

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

Example

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

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 *)

Crossrefs

Cf. A283962.

Sequence in context: A191711 A163280 A056537 * A359298 A255127 A083221

Adjacent sequences: A293051 A293052 A293053 * A293055 A293056 A293057

Keyword

nonn,easy,tabl

Author

Clark Kimberling, Oct 06 2017

Status

approved