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A191454
Dispersion of (2*floor(n*r)), where r=(golden ratio), by antidiagonals.
1
1, 2, 3, 6, 8, 4, 18, 24, 12, 5, 58, 76, 38, 16, 7, 186, 244, 122, 50, 22, 9, 600, 788, 394, 160, 70, 28, 10, 1940, 2550, 1274, 516, 226, 90, 32, 11, 6276, 8250, 4122, 1668, 730, 290, 102, 34, 13, 20308, 26696, 13338, 5396, 2362, 938, 330, 110, 42, 14, 65718
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.
EXAMPLE
Northwest corner:
1...2....6....18...58
3...8....24...76...244
4...12...38...122..394
5...16...50...160..516
7...22...70...226..730
MATHEMATICA
(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; x=GoldenRatio;
f[n_] :=2Floor[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}]]
(* A191454 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191454 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)
CROSSREFS
KEYWORD
nonn,tabl,changed
AUTHOR
Clark Kimberling, Jun 05 2011
STATUS
approved