A191440 - OEIS

%I #12 Oct 21 2024 00:46:51

%S 1,3,2,8,6,4,20,15,11,5,49,37,28,13,7,119,90,69,32,18,9,288,218,168,

%T 78,44,23,10,696,527,407,189,107,57,25,12,1681,1273,984,457,259,139,

%U 61,30,14,4059,3074,2377,1104,626,337,148,73,35,16,9800,7422,5740

%N Dispersion of ([n*sqrt(2)+n+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....3....8....20...49

%e   2....6....15...37...90

%e   4....11...28...69...168

%e   5....13...32...78...189

%e   7....18...44...107..259

%t (* Program generates the dispersion array T of increasing sequence f[n] *)

%t r=40; r1=12; c=40; c1=12;  x = Sqr[2];

%t f[n_] := Floor[n*x+n+3/2] (* complement of column 1 *)

%t mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, 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]];

%t TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]

%t (* A191440 array *)

%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191440 sequence *)

%t (* Program by _Peter J. C. Moses_, Jun 01 2011 *)

%Y Cf. A114537, A035513, A035506, A191438, A191439, A083087.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Jun 04 2011