

A191539


Dispersion of (5nfloor(n*sqrt(5))), by antidiagonals.


1



1, 3, 2, 9, 6, 4, 25, 17, 12, 5, 70, 47, 34, 14, 7, 194, 130, 94, 39, 20, 8, 537, 360, 260, 108, 56, 23, 10, 1485, 996, 719, 299, 155, 64, 28, 11, 4105, 2753, 1988, 827, 429, 177, 78, 31, 13, 11346, 7610, 5495, 2286, 1186, 490, 216, 86, 36, 15, 31360, 21034
(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 nth 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: A191426A191455 and A191536A191545.


LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened


EXAMPLE

Northwest corner:
1...3....9....25...70
2...6....17...47...130
4...12...34...94...260
5...14...39..108...299
7...20...56...155..429


MATHEMATICA

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; f[n_] :=5nFloor[n*Sqrt[5]] (* 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, r1}, {j, 1, c1}]] (* A191539 array *)
Flatten[Table[t[k, n  k + 1], {n, 1, c1}, {k, 1, n}]] (* A191539 sequence *)


CROSSREFS

Cf. A114537, A035513, A035506.
Sequence in context: A329211 A164279 A289053 * A235539 A191449 A175840
Adjacent sequences: A191536 A191537 A191538 * A191540 A191541 A191542


KEYWORD

nonn,tabl


AUTHOR

Clark Kimberling, Jun 06 2011


STATUS

approved



