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A191431 Dispersion of ([nx+x]), where x=sqrt(2) and [ ]=floor, by antidiagonals. 1
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 13, 16, 18, 21, 22, 19, 17, 24, 26, 31, 32, 28, 25, 20, 35, 38, 45, 46, 41, 36, 29, 23, 50, 55, 65, 66, 59, 52, 42, 33, 27, 72, 79, 93, 94, 84, 74, 60, 48, 39, 30, 103, 113, 132, 134, 120, 106, 86, 69, 56, 43, 34, 147, 161, 188, 190, 171, 151, 123, 98, 80, 62, 49, 37 (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.....2....4....7...11...16

3.....5....8...12...18...26

6.....9...14...21...31...45

10...15...22...32...46...66

13...19...28...41...59...84

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 = Sqrt[2];

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}]]

(* A191431 array *)

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

(* A191431 sequence *)

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

CROSSREFS

Cf. A114537, A035513, A035506.

Sequence in context: A044915 A161951 A290492 * A191430 A011760 A085265

Adjacent sequences:  A191428 A191429 A191430 * A191432 A191433 A191434

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jun 03 2011

STATUS

approved

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Last modified April 16 17:01 EDT 2021. Contains 343050 sequences. (Running on oeis4.)