A332502 Rectangular array read by antidiagonals: T(n,k) = floor(n + k*r), where r = golden ratio = (1+sqrt(5))/2.

0, 1, 1, 3, 2, 2, 4, 4, 3, 3, 6, 5, 5, 4, 4, 8, 7, 6, 6, 5, 5, 9, 9, 8, 7, 7, 6, 6, 11, 10, 10, 9, 8, 8, 7, 7, 12, 12, 11, 11, 10, 9, 9, 8, 8, 14, 13, 13, 12, 12, 11, 10, 10, 9, 9, 16, 15, 14, 14, 13, 13, 12, 11, 11, 10, 10, 17, 17, 16, 15, 15, 14, 14, 13

Offset: 0

Author

Clark Kimberling, May 08 2020

Keywords

• nonn
• tabl
• easy

Comments

Every nonnegative integer occurs exactly once in the union of row 0 and the main diagonal.

Column 0: Nonnegative integers, A001477.

Row 0: Lower Wythoff sequence, A000201.

Row 1: A026351.

Row 2: A026355 (and essentially A099267).

Main Diagonal: Upper Wythoff sequence, A001950.

Diagonal (1,4,6,9,...) = A003622;

Diagonal (3,5,8,11,...) = A026274;

Diagonal (1,3,6,8,...) = A026352;

Diagonal (2,4,7,9,...) = A026356.

Examples

			Northwest corner:
  0   1   3   4   6   8    9    11   12   14   16
  1   2   4   5   7   9    10   12   13   15   17
  2   3   5   6   8   10   11   13   14   16   18
  3   4   6   7   9   11   12   14   15   17   19
  4   5   7   8   10  12   13   15   16   18   20
  5   6   8   9   11  13   14   16   17   19   21
As a triangle (antidiagonals):
  0
  1   1
  2   2   3
  3   3   4   4
  4   4   5   5   6
  5   5   6   6   7   8
  6   6   7   7   8   9   9
		

Crossrefs

Cf. A000210, A001950, A001622, A084531, A167267, A019446.

Programs

Mathematica

t[n_, k_] := Floor[n + k*GoldenRatio];
Grid[Table[t[n, k], {n, 0, 10}, {k, 0, 10}]] (* array *)
u = Table[t[n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten  (* sequence *)

Formula

T(n,k) = floor(n + k*r), where r = golden ratio = (1+sqrt(5))/2.