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A191434 Dispersion of ([n*x+n+3/2]), where x=(golden ratio) and [ ]=floor, by antidiagonals.

%I A191434 #14 Oct 20 2024 16:09:51
%S A191434 1,4,2,11,6,3,30,17,9,5,80,46,25,14,7,210,121,66,38,19,8,551,318,174,
%T A191434 100,51,22,10,1444,834,457,263,135,59,27,12,3781,2184,1197,690,354,
%U A191434 155,72,32,13,9900,5719,3135,1807,928,407,189,85,35,15,25920,14974
%N A191434 Dispersion of ([n*x+n+3/2]), where x=(golden ratio) and [ ]=floor, by antidiagonals.
%C A191434 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 A191434 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191434 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191434 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191434 More recent examples of dispersions: A191426-A191455.
%e A191434 Northwest corner:
%e A191434   1.....4....11....30...80
%e A191434   2.....6....17....46...121
%e A191434   3.....9....25....66...174
%e A191434   5.....14...38...100...263
%e A191434   7.....19...51...135...354
%t A191434 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191434 r = 40; r1 = 12; c = 40; c1 = 12;
%t A191434 x = 1 + GoldenRatio; f[n_] := Floor[n*x + 3/2]
%t A191434 (* f(n) is complement of column 1 *)
%t A191434 mex[list_] :=  NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,
%t A191434   Length[Union[list]]]
%t A191434 rows = {NestList[f, 1, c]};
%t A191434 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191434 t[i_, j_] := rows[[i, j]];
%t A191434 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191434 (* A191434 array *)
%t A191434 Flatten[Table[
%t A191434   t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
%t A191434 (* A191434 sequence *)
%t A191434 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191434 Cf. A114537, A035513, A035506, A191426.
%K A191434 nonn,tabl
%O A191434 1,2
%A A191434 _Clark Kimberling_, Jun 04 2011