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A191427 Dispersion of ([n*r+3/2]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.

%I A191427 #17 Oct 20 2024 21:02:09
%S A191427 1,3,2,6,4,5,11,7,9,8,19,12,16,14,10,32,20,27,24,17,13,53,33,45,40,29,
%T A191427 22,15,87,54,74,66,48,37,25,18,142,88,121,108,79,61,41,30,21,231,143,
%U A191427 197,176,129,100,67,50,35,23,375,232,320,286,210,163,109,82,58,38,26,608,376,519,464,341,265,177,134,95,62,43,28
%N A191427 Dispersion of ([n*r+3/2]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.
%C A191427 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 A191427 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191427 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191427 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191427 More recent examples of dispersions: A191426-A191455.
%e A191427 Northwest corner:
%e A191427   1...3...6...11..19
%e A191427   2...4...7...12..20
%e A191427   5...9...16..27..45
%e A191427   8...14..24..40..66
%e A191427   10..17..29..48..79
%t A191427 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191427 r = 40; r1 = 12; (* r=#rows of T, r1=#rows to show *)
%t A191427 c = 40; c1 = 12; (* c=#cols of T, c1=#cols to show *)
%t A191427 x = GoldenRatio; f[n_] := Floor[n*x + 3/2]
%t A191427 mex[list_] :=
%t A191427 NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,
%t A191427   Length[Union[list]]]
%t A191427 rows = {NestList[f, 1, c]};
%t A191427 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191427 t[i_, j_] := rows[[i, j]];
%t A191427 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191427 (* A191427 array *)
%t A191427 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
%t A191427 (* A191427 sequence *)
%t A191427 (* _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191427 Cf. A114537, A035513, A035506.
%K A191427 nonn,tabl
%O A191427 1,2
%A A191427 _Clark Kimberling_, Jun 02 2011