A333029 Order array of array A333028, read by antidiagonals.

1, 2, 4, 3, 6, 10, 5, 8, 15, 11, 7, 12, 24, 18, 13, 9, 19, 38, 28, 21, 16, 14, 30, 58, 45, 32, 25, 17, 22, 47, 77, 65, 50, 40, 27, 20, 35, 67, 95, 84, 70, 60, 44, 31, 23, 55, 86, 112, 102, 89, 79, 64, 49, 36, 26, 74, 104, 128, 118, 106, 97, 83, 69, 56, 43

Offset: 1

Author

Clark Kimberling, Mar 10 2020

Keywords

• nonn
• tabl

Comments

Suppose that (r(n,k)), for n >=1 and k >= 1 is an array of distinct numbers. Replace each r(n,k) by its position when all the numbers r(n,k) are ordered by <. The resulting array is the order array of (r(n,k)). The array A333029 is an interspersion and, as a sequence, a permutation of the natural numbers.

Examples

			Northwest corner:
   1   2   3   5   7    9   14   22
   4   6   8  12  19   30   47   67
  10  15  24  38  58   77   95  112
  11  18  24  45  65   84  102  118
  13  21  32  50  70   89  106  122
  16  25  40  60  79   97  114  130
  17  27  44  64  83  101  117  132
		

References

• Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168.

Links

• Clark Kimberling, Lucas Representations of Positive Integers, J. Int. Seq., Vol. 23 (2020), Article 20.9.5.

Crossrefs

Cf. A333028.

Programs

Mathematica

W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
t = Table[GCD[W[n, 1], W[n, 2]], {n, 1, 500}];
u = Flatten[Position[t, 1]] ; v[n_, k_] := W[u[[n]], k];
g = Sort[Table[v[n - k + 1, k], {n, 20}, {k, n, 1, -1}] // Flatten]
wo[n_, k_] := Length[Intersection[Range[v[n, k]], Complement[Range[1500], g]]]
WP[n_, k_] := v[n, k] - wo[n, k];
TableForm[Table[WP[n, k], {n, 1, 15}, {k, 1, 10}]]  (* A333029 array *)
Table[WP[n - k + 1, k], {n, 16}, {k, n, 1, -1}] // Flatten  (* A333029 sequence *)