A175004 Interspersion related to the Wythoff Array.

1, 2, 3, 4, 6, 5, 7, 10, 9, 8, 12, 17, 15, 14, 11, 20, 28, 25, 23, 19, 13, 33, 46, 41, 38, 31, 22, 16, 54, 75, 67, 62, 51, 36, 27, 18, 88, 122, 109, 101, 83, 59, 44, 30, 21, 143, 198, 177, 164, 135, 96, 72, 49, 35, 24, 232, 321, 287, 266, 219, 156, 117, 80, 57, 40, 26, 376, 520, 465, 431, 355, 253, 190, 130, 93, 65, 43, 29

Offset: 1

Author

Clark Kimberling, Apr 03 2010

Keywords

• nonn
• tabl

Comments

The rows satisfy the recurrence r(n)=r(n-1)+r(n-2)+1.

Every positive integer occurs exactly once, so that as a sequence, A175004 is a permutation of the natural numbers. As an array, it is an interspersion, hence also a dispersion. Specifically, it is the dispersion of the sequence floor(n*x+2/x), where x=(golden ratio). For a discussion of dispersions, see A191426.

Examples

			Corner of the array:
1....2....4....7....12...20... (cf. A000071)
3....6....10...17...28...46... (cf. A001610)
5....9....15...25...41...67... (cf. A001595)
8....14...23...38...62...101..
		

Crossrefs

Cf. A035513.

Programs

Mathematica

(* program generates the dispersion array T of the complement of increasing sequence f[n] *)
r = 40; r1 = 12; (* r=# rows of T, r1=# rows to show *)
c = 40; c1 = 12; (* c= # cols of T, c1=# cols to show *)
x = GoldenRatio; f[n_] := Floor[n*x + 2/x]
(* f(n) is complement of column 1 *)
mex[list_] :=  NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A175004 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* array as a sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011, added here Jun 03 2011 by Clark Kimberling *)

Formula

Let W'=W-1, where W is the Wythoff array, given by A035513.

Row 1 of W' is (0,1,2,4,7,12,...); replace this by (1,2,4,7,12,...).

The resulting array is A175004.