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A191441 Dispersion of ([n*x+n+x]), where x=sqrt(2) and [ ]=floor, by antidiagonals. 1
1, 4, 2, 12, 7, 3, 31, 19, 9, 5, 77, 48, 24, 14, 6, 188, 118, 60, 36, 16, 8, 456, 287, 147, 89, 41, 21, 10, 1103, 695, 357, 217, 101, 53, 26, 11, 2665, 1680, 864, 526, 246, 130, 65, 28, 13, 6436, 4058, 2088, 1272, 596, 316, 159, 70, 33, 15, 15540, 9799, 5043 (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..58.

EXAMPLE

Northwest corner:

1....4....12...31...77

2....7....19...48...118

3....9....24...60...147

5....14...36...89...217

6....16...41...101..246

MATHEMATICA

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

r=40; r1=12; c=40; c1=12;  x = Sqr[2];

f[n_] := Floor[n*x+n+x] (* 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}]]

(* A191441 array *)

Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191441 sequence *)

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

CROSSREFS

Cf. A114537, A035513, A035506, A191438.

Sequence in context: A293603 A201825 A104007 * A152664 A167591 A227043

Adjacent sequences:  A191438 A191439 A191440 * A191442 A191443 A191444

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jun 04 2011

STATUS

approved

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Last modified February 25 04:32 EST 2020. Contains 332217 sequences. (Running on oeis4.)