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A191433 Dispersion of ([nx+n+1/2]), where x=(golden ratio) and [ ]=floor, by antidiagonals. 1
1, 3, 2, 8, 5, 4, 21, 13, 10, 6, 55, 34, 26, 16, 7, 144, 89, 68, 42, 18, 9, 377, 233, 178, 110, 47, 24, 11, 987, 610, 466, 288, 123, 63, 29, 12, 2584, 1597, 1220, 754, 322, 165, 76, 31, 14, 6765, 4181, 3194, 1974, 843, 432, 199, 81, 37, 15, 17711, 10946 (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....3....8....21...55...144

2....5....13...34...89...233

4....10...26...68...178..466

6....16...42...110..288..754

7....18...47...123..322..843

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 = 1 + GoldenRatio;

f[n_] := Floor[n*x + 1/2] (* 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}]] (* A191433 array *)

Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191433 sequence *)

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

CROSSREFS

Cf. A114537, A035513, A035506.

Sequence in context: A271589 A083087 A191724 * A191437 A072788 A085178

Adjacent sequences:  A191430 A191431 A191432 * A191434 A191435 A191436

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jun 03 2011

STATUS

approved

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Last modified November 21 16:04 EST 2019. Contains 329371 sequences. (Running on oeis4.)