|
%I #11 Feb 14 2014 00:28:16
%S 1,2,3,4,5,6,7,8,9,10,11,12,14,15,13,17,18,21,22,19,16,25,26,31,32,28,
%T 24,20,36,38,45,46,41,35,29,23,52,55,65,66,59,50,42,34,27,75,79,93,94,
%U 84,72,60,49,39,30,107,113,133,134,120,103,86,70,56,43,33,152,161,189,191,171,147,123,100,80,62,48,37
%N Dispersion of ([n*sqrt(2)+3/2]), where [ ]=floor, by antidiagonals.
%C 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:
%C (1) s=A000040 (the primes), D=A114537, u=A114538.
%C (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C More recent examples of dispersions: A191426-A191455.
%e Northwest corner:
%e 1...2...4...7..11
%e 3...5...12...18..18
%e 6...9...14..21..31
%e 10...15..22..32..46
%e 13..19..28..41..59
%t (* Program generates the dispersion array T of increasing sequence f[n] *)
%t r = 40; r1 = 12; (* r=# rows of T to compute, r1=# rows to show *)
%t c = 40; c1 = 12; (* c=# cols to compute, c1=# cols to show *)
%t x = Sqrt[2];
%t f[n_] := Floor[n*x + 3/2] (* f(n) is complement of column 1 *)
%t mex[list_] :=
%t NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,
%t Length[Union[list]]]
%t rows = {NestList[f, 1, c]};
%t Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t t[i_, j_] := rows[[i, j]]; (* the array T *)
%t TableForm[
%t Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191430 array *)
%t Flatten[Table[
%t t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191430 sequence *)
%t (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y Cf. A114537, A035513, A035506.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Jun 03 2011
|
