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A191436 Dispersion of ([n*x+n+x-1]), where x=(golden ratio) and [ ]=floor, by antidiagonals.

%I A191436 #12 Oct 21 2024 00:46:46
%S A191436 1,4,2,12,6,3,33,17,9,5,88,46,25,14,7,232,122,67,38,19,8,609,321,177,
%T A191436 101,51,22,10,1596,842,465,266,135,59,27,11,4180,2206,1219,698,355,
%U A191436 156,72,30,13,10945,5777,3193,1829,931,410,190,80,35,15,28656,15126
%N A191436 Dispersion of ([n*x+n+x-1]), where x=(golden ratio) and [ ]=floor, by antidiagonals.
%C A191436 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 A191436 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191436 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191436 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191436 More recent examples of dispersions: A191426-A191455.
%e A191436 Northwest corner:
%e A191436   1....4....12...33...88
%e A191436   2....6....17...46...122
%e A191436   3....9....25...67...177
%e A191436   5....14...38...101..266
%e A191436   7....19...51...135..355
%t A191436 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191436 r = 40; r1 = 12;  c = 40; c1 = 12;  x = GoldenRatio;
%t A191436 f[n_] := Floor[n*x+n+x-1] (* complement of column 1 *)
%t A191436 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191436 rows = {NestList[f, 1, c]};
%t A191436 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191436 t[i_, j_] := rows[[i, j]];
%t A191436 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191436 (* A191436 array *)
%t A191436 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191436 sequence *)
%t A191436 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191436 Cf. A114537, A035513, A035506.
%K A191436 nonn,tabl
%O A191436 1,2
%A A191436 _Clark Kimberling_, Jun 04 2011