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A191426 Dispersion of (3+[n*r]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.

%I A191426 #39 Oct 20 2024 21:01:59
%S A191426 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 A191426 19,13,140,106,72,59,46,33,24,16,229,174,119,98,77,56,41,28,18,373,
%U A191426 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 A191426 Dispersion of (3+[n*r]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.
%C A191426 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 A191426 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191426 (2) s=A022342 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191426 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191426 More recent examples of dispersions: A191426-A191455.
%D A191426 Clark Kimberling, Fractal sequences and interspersions, Ars Combinatoria 45 (1997) 157-168.
%H A191426 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 A191426 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 A191426 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 A191426 Northwest corner:
%e A191426   1...4...9...17..30
%e A191426   2...6...12..22..38
%e A191426   3...7...14..25..43
%e A191426   5...11..20..35..59
%e A191426   8...15..27..46..77
%t A191426 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191426 r = 40; r1 = 12; (* r=#rows of T, r1=#rows to show *)
%t A191426 c = 40; c1 = 12; (* c=#cols of T, c1=#cols to show *)
%t A191426 x = GoldenRatio; f[n_] := Floor[n*x + 3]
%t A191426 mex[list_] :=  NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191426 rows = {NestList[f, 1, c]};
%t A191426 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191426 t[i_, j_] := rows[[i, j]];
%t A191426 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191426 (* A191426 array *)
%t A191426 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]  (* A191426 sequence *)
%t A191426 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191426 Cf. A114537, A035513, A035506.
%K A191426 nonn,tabl
%O A191426 1,2
%A A191426 _Clark Kimberling_, Jun 02 2011