cp's OEIS Frontend

A191428 Dispersion of ([n*r+r]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.

%I A191428 #12 Oct 20 2024 21:02:21
%S A191428 1,3,2,6,4,5,11,8,9,7,19,14,16,12,10,32,24,27,21,17,13,53,40,45,35,29,
%T A191428 22,15,87,66,74,58,48,37,25,18,142,108,121,95,79,61,42,30,20,231,176,
%U A191428 197,155,129,100,69,50,33,23,375,286,320,252,210,163,113,82,55,38,26,608,464,519,409,341,265,184,134,90,63,43,28
%N A191428 Dispersion of ([n*r+r]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.
%C A191428 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 A191428 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191428 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191428 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191428 More recent examples of dispersions: A191426-A191455.
%e A191428 Northwest corner:
%e A191428   1...3...6...11..19
%e A191428   2...4...8...14..24
%e A191428   5...9...16..27..45
%e A191428   7...12..21..35..58
%e A191428   10..17..29..48..79
%t A191428 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191428 r = 40; r1 = 12;  (* r=# rows of T, r1=# rows to show *)
%t A191428 c = 40; c1 = 12;  (* c=# cols of T, c1=# cols to show *)
%t A191428 x = GoldenRatio; f[n_] := Floor[n*x + x]
%t A191428 (* f(n) is complement of column 1 *)
%t A191428 mex[list_] :=
%t A191428 NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,
%t A191428   Length[Union[list]]]
%t A191428 rows = {NestList[f, 1, c]};
%t A191428 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191428 t[i_, j_] := rows[[i, j]];
%t A191428 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191428 (* A191428 array *)
%t A191428 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
%t A191428 (* A191428 sequence *)
%t A191428 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191428 Cf. A114537, A035513, A035506.
%K A191428 nonn,tabl
%O A191428 1,2
%A A191428 _Clark Kimberling_, Jun 03 2011