

A191536


Dispersion of (3+floor(n*sqrt(2))), by antidiagonals.


5



1, 4, 2, 8, 5, 3, 14, 10, 7, 6, 22, 17, 12, 11, 9, 34, 27, 19, 18, 15, 13, 51, 41, 29, 28, 24, 21, 16, 75, 60, 44, 42, 36, 32, 25, 20, 109, 87, 65, 62, 53, 48, 38, 31, 23, 157, 126, 94, 90, 77, 70, 56, 46, 35, 26, 225, 181, 135, 130, 111, 101, 82, 68, 52, 39
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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...4....8....14...22
2...5....10...17...27
3...7....12...19...29
6...11...18...28...42
9...15...24...36...54


MATHEMATICA

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; f[n_] :=3+Floor[n*Sqrt[2]] (* 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}]]
(* A191536 array *)
Flatten[Table[t[k, n  k + 1], {n, 1, c1}, {k, 1, n}]] (* A191536 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)


CROSSREFS

Cf. A114537, A035513, A035506.
Sequence in context: A131819 A194064 A194054 * A187076 A000727 A030181
Adjacent sequences: A191533 A191534 A191535 * A191537 A191538 A191539


KEYWORD

nonn,tabl


AUTHOR

Clark Kimberling, Jun 06 2011


STATUS

approved



