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A191433 Dispersion of ([n*x+n+1/2]), where x=(golden ratio) and [ ]=floor, by antidiagonals.

%I A191433 #13 Oct 20 2024 21:03:01
%S A191433 1,3,2,8,5,4,21,13,10,6,55,34,26,16,7,144,89,68,42,18,9,377,233,178,
%T A191433 110,47,24,11,987,610,466,288,123,63,29,12,2584,1597,1220,754,322,165,
%U A191433 76,31,14,6765,4181,3194,1974,843,432,199,81,37,15,17711,10946
%N A191433 Dispersion of ([n*x+n+1/2]), where x=(golden ratio) and [ ]=floor, by antidiagonals.
%C A191433 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 A191433 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191433 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191433 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191433 More recent examples of dispersions: A191426-A191455.
%e A191433 Northwest corner:
%e A191433   1....3....8....21...55...144
%e A191433   2....5....13...34...89...233
%e A191433   4....10...26...68...178..466
%e A191433   6....16...42...110..288..754
%e A191433   7....18...47...123..322..843
%t A191433 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191433 r = 40; r1 = 12;  (* r=# rows of T, r1=# rows to show *)
%t A191433 c = 40; c1 = 12;  (* c=# cols of T, c1=# cols to show *)
%t A191433 x = 1 + GoldenRatio;
%t A191433 f[n_] := Floor[n*x + 1/2] (* f(n) is complement of column 1 *)
%t A191433 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,
%t A191433   Length[Union[list]]]
%t A191433 rows = {NestList[f, 1, c]};
%t A191433 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191433 t[i_, j_] := rows[[i, j]];
%t A191433 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191433 array *)
%t A191433 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191433 sequence *)
%t A191433 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191433 Cf. A114537, A035513, A035506.
%K A191433 nonn,tabl
%O A191433 1,2
%A A191433 _Clark Kimberling_, Jun 03 2011