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A191434 Dispersion of ([nx+n+3/2]), where x=(golden ratio) and [ ]=floor, by antidiagonals. 1
1, 4, 2, 11, 6, 3, 30, 17, 9, 5, 80, 46, 25, 14, 7, 210, 121, 66, 38, 19, 8, 551, 318, 174, 100, 51, 22, 10, 1444, 834, 457, 263, 135, 59, 27, 12, 3781, 2184, 1197, 690, 354, 155, 72, 32, 13, 9900, 5719, 3135, 1807, 928, 407, 189, 85, 35, 15, 25920, 14974 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n)), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:

(1) s=A000040 (the primes), D=A114537, u=A114538.

(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.

(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.

More recent examples of dispersions: A191426-A191455.

LINKS

Table of n, a(n) for n=1..57.

EXAMPLE

Northwest corner:

1.....4....11....30...80

2.....6....17....46...121

3.....9....25....66...174

5.....14...38...100...263

7.....19...51...135...354

MATHEMATICA

(* Program generates the dispersion array T of increasing sequence f[n] *)

r = 40; r1 = 12; c = 40; c1 = 12;

x = 1 + GoldenRatio; f[n_] := Floor[n*x + 3/2]

(* 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}]]

(* A191434 array *)

Flatten[Table[

  t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]

(* A191434 sequence *)

(* Program by Peter J. C. Moses, Jun 01 2011 *)

CROSSREFS

Cf. A114537, A035513, A035506, A191426.

Sequence in context: A091484 A163544 A191728 * A163918 A318784 A154699

Adjacent sequences:  A191431 A191432 A191433 * A191435 A191436 A191437

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jun 04 2011

STATUS

approved

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Last modified April 4 05:34 EDT 2020. Contains 333212 sequences. (Running on oeis4.)