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A175004 Interspersion related to the Wythoff Array. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..78.

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.

EXAMPLE

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..

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 *)

CROSSREFS

Cf. A035513.

Sequence in context: A194981 A057027 A090894 * A191734 A191443 A136175

Adjacent sequences:  A175001 A175002 A175003 * A175005 A175006 A175007

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Apr 03 2010

STATUS

approved

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Last modified July 6 23:56 EDT 2020. Contains 335484 sequences. (Running on oeis4.)