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

1,2

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.

References

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

Links

Table of n, a(n) for n=1..65.

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

Example

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

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 *)

Crossrefs

Cf. A333028.

Sequence in context: A113233 A051849 A283961 * A175498 A318452 A083673

Adjacent sequences: A333026 A333027 A333028 * A333030 A333031 A333032

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 10 2020

Status

approved