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A191447 Dispersion of (1+[n*sqrt(5)]), where [ ]=floor, by antidiagonals.

%I A191447 #16 Oct 21 2024 00:49:39
%S A191447 1,3,2,7,5,4,16,12,9,6,36,27,21,14,8,81,61,47,32,18,10,182,137,106,72,
%T A191447 41,23,11,407,307,238,161,92,52,25,13,911,687,533,361,206,117,56,30,
%U A191447 15,2038,1537,1192,808,461,262,126,68,34,17,4558,3437,2666,1807
%N A191447 Dispersion of (1+[n*sqrt(5)]), where [ ]=floor, by antidiagonals.
%C A191447 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 A191447 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191447 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191447 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191447 More recent examples of dispersions: A191426-A191455.
%e A191447 Northwest corner:
%e A191447   1...3....7....16...36
%e A191447   2...5....12...27...61
%e A191447   4...9....21...47...106
%e A191447   6...14...32...72...161
%e A191447   8...18...41...92...206
%t A191447 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191447 r=40; r1=12; c=40; c1=12;  x = Sqrt[5];
%t A191447 f[n_] := 1+Floor[n*x] (* complement of column 1 *)
%t A191447 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191447 rows = {NestList[f, 1, c]};
%t A191447 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191447 t[i_, j_] := rows[[i, j]];
%t A191447 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191447 (* A191447 array *)
%t A191447 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191447 sequence *)
%t A191447 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191447 Cf. A114537, A035513, A035506, A191446.
%K A191447 nonn,tabl
%O A191447 1,2
%A A191447 _Clark Kimberling_, Jun 05 2011