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A191446 Dispersion of [n*sqrt(5)], where [ ]=floor, by antidiagonals.

%I A191446 #18 Oct 21 2024 00:49:43
%S A191446 1,2,3,4,6,5,8,13,11,7,17,29,24,15,9,38,64,53,33,20,10,84,143,118,73,
%T A191446 44,22,12,187,319,263,163,98,49,26,14,418,713,588,364,219,109,58,31,
%U A191446 16,934,1594,1314,813,489,243,129,69,35,18,2088,3564,2938,1817
%N A191446 Dispersion of [n*sqrt(5)], where [ ]=floor, by antidiagonals.
%C A191446 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 A191446 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191446 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191446 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191446 More recent examples of dispersions: A191426-A191455.
%e A191446 Northwest corner:
%e A191446   1...2....4....8...17
%e A191446   3...6....13...29..64
%e A191446   5...11...24...53..118
%e A191446   7...15...33...73..163
%e A191446   9...20...44...98..219
%t A191446 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191446 r=40; r1=12; c=40; c1=12;  x = Sqrt[5];
%t A191446 f[n_] := Floor[n*x] (* complement of column 1 *)
%t A191446 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191446 rows = {NestList[f, 1, c]};
%t A191446 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191446 t[i_, j_] := rows[[i, j]];
%t A191446 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191446 (* A191446 array *)
%t A191446 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191446 sequence *)
%t A191446 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191446 Cf. A114537, A035513, A035506.
%K A191446 nonn,tabl
%O A191446 1,2
%A A191446 _Clark Kimberling_, Jun 05 2011
%E A191446 Corrected typo in name and fixed Mathematica program by _Vaclav Kotesovec_, Oct 24 2014