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A191429 Dispersion of ([n*sqrt(2)+2]), where [ ]=floor, by antidiagonals. 2
1, 3, 2, 6, 4, 5, 10, 7, 9, 8, 16, 11, 14, 13, 12, 24, 17, 21, 20, 18, 15, 35, 26, 31, 30, 27, 23, 19, 51, 38, 45, 44, 40, 34, 28, 22, 74, 55, 65, 64, 58, 50, 41, 33, 25, 106, 79, 93, 92, 84, 72, 59, 48, 37, 29, 151, 113, 133, 132, 120, 103, 85, 69, 54, 43, 32, 215, 161, 190, 188, 171, 147, 122, 99, 78, 62, 47, 36 (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..78.

EXAMPLE

Northwest corner:

1...3...6...10..16

2...4...7...11..17

5...9...14..21..31

8...13..20..30..44

12..18..27..40..58

MATHEMATICA

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

r = 40; r1 = 12;  (* r=# rows of T to compute, r1=# rows to show *)

c = 40; c1 = 12;   (* c=# cols to compute, c1=# cols to show *)

x = Sqrt[2];

f[n_] := Floor[n*x + 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]];  (* the array T *)

TableForm[

Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191429 array *)

Flatten[Table[

  t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191429 sequence *)

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

CROSSREFS

Cf. A114537, A035513, A035506.

Sequence in context: A194918 A194915 A195077 * A191655 A120911 A064789

Adjacent sequences:  A191426 A191427 A191428 * A191430 A191431 A191432

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jun 03 2011

STATUS

approved

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Last modified November 30 05:12 EST 2021. Contains 349419 sequences. (Running on oeis4.)