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A191428 Dispersion of ([nr+r]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals. 1
1, 3, 2, 6, 4, 5, 11, 8, 9, 7, 19, 14, 16, 12, 10, 32, 24, 27, 21, 17, 13, 53, 40, 45, 35, 29, 22, 15, 87, 66, 74, 58, 48, 37, 25, 18, 142, 108, 121, 95, 79, 61, 42, 30, 20, 231, 176, 197, 155, 129, 100, 69, 50, 33, 23, 375, 286, 320, 252, 210, 163, 113, 82, 55, 38, 26, 608, 464, 519, 409, 341, 265, 184, 134, 90, 63, 43, 28 (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...11..19

2...4...8...14..24

5...9...16..27..45

7...12..21..35..58

10..17..29..48..79

MATHEMATICA

(* Program generates the dispersion array T 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 + 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}]]

(* A191428 array *)

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

(* A191428 sequence *)

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

CROSSREFS

Cf. A114537, A035513, A035506.

Sequence in context: A274315 A194059 A191427 * A191733 A191444 A258241

Adjacent sequences:  A191425 A191426 A191427 * A191429 A191430 A191431

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jun 03 2011

STATUS

approved

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Last modified January 20 21:46 EST 2022. Contains 350472 sequences. (Running on oeis4.)