

A191428


Dispersion of ([nr+r]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.


1



1, 3, 2, 6, 4, 5, 11, 8, 9, 7, 19, 14, 16, 12, 10, 32, 24, 27, 21, 17, 13, 53, 40, 45, 35, 29, 22, 15, 87, 66, 74, 58, 48, 37, 25, 18, 142, 108, 121, 95, 79, 61, 42, 30, 20, 231, 176, 197, 155, 129, 100, 69, 50, 33, 23, 375, 286, 320, 252, 210, 163, 113, 82, 55, 38, 26, 608, 464, 519, 409, 341, 265, 184, 134, 90, 63, 43, 28
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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.


LINKS

Table of n, a(n) for n=1..78.


EXAMPLE

Northwest corner:
1...3...6...11..19
2...4...8...14..24
5...9...16..27..45
7...12..21..35..58
10..17..29..48..79


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 = GoldenRatio; 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}]]
(* A191428 array *)
Flatten[Table[t[k, n  k + 1], {n, 1, c1}, {k, 1, n}]]
(* A191428 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)


CROSSREFS

Cf. A114537, A035513, A035506.
Sequence in context: A274315 A194059 A191427 * A191733 A191444 A258241
Adjacent sequences: A191425 A191426 A191427 * A191429 A191430 A191431


KEYWORD

nonn,tabl


AUTHOR

Clark Kimberling, Jun 03 2011


STATUS

approved



