A191426 - OEIS

%I #39 Oct 20 2024 21:01:59

%S 1,4,2,9,6,3,17,12,7,5,30,22,14,11,8,51,38,25,20,15,10,85,64,43,35,27,

%T 19,13,140,106,72,59,46,33,24,16,229,174,119,98,77,56,41,28,18,373,

%U 284,195,161,127,93,69,48,32,21,606,462,318,263,208,153,114,80,54,36,23,983,750,517,428,339,250,187,132,90,61,40,26

%N Dispersion of (3+[n*r]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=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=A022342 (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.

%D Clark Kimberling, Fractal sequences and interspersions, Ars Combinatoria 45 (1997) 157-168.

%H Mohammad K. Azarian, <a href="http://www.math-cs.ucmo.edu/~mjms/1998.3/prob.ps">Problem 123</a>, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3, Fall 1998, p. 176. <a href="http://www.math-cs.ucmo.edu/~mjms/2000.1/soln.ps">Solution</a> published in Vol. 12, No. 1, Winter 2000, pp. 61-62.

%H Clark Kimberling, <a href="http://dx.doi.org/10.1090/S0002-9939-1993-1111434-0">Interspersions and dispersions</a>, Proceedings of the American Mathematical Society, 117 (1993) 313-321.

%H A. J. Macfarlane, <a href="https://arxiv.org/abs/2405.18128">On the fibbinary numbers and the Wythoff array</a>, arXiv:2405.18128 [math.CO], 2024. See page 2.

%e Northwest corner:

%e   1...4...9...17..30

%e   2...6...12..22..38

%e   3...7...14..25..43

%e   5...11..20..35..59

%e   8...15..27..46..77

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

%t r = 40; r1 = 12; (* r=#rows of T, r1=#rows to show *)

%t c = 40; c1 = 12; (* c=#cols of T, c1=#cols to show *)

%t x = GoldenRatio; f[n_] := Floor[n*x + 3]

%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 (* A191426 array *)

%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]  (* A191426 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 02 2011