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A191432 Dispersion of ([n*x+1/x]), where x=sqrt(2) and [ ]=floor, by antidiagonals.

%I A191432 #29 Oct 20 2024 16:22:35
%S A191432 1,2,5,3,7,8,4,10,12,11,6,14,17,16,15,9,20,24,23,21,18,13,28,34,33,30,
%T A191432 26,22,19,40,48,47,43,37,31,25,27,57,68,67,61,53,44,36,29,38,81,96,95,
%U A191432 86,75,62,51,41,32,54,115,136,135,122,106,88,72,58,45,35,77,163,193,191,173,150,125,102,82,64,50,39
%N A191432 Dispersion of ([n*x+1/x]), where x=sqrt(2) and [ ]=floor, by antidiagonals.
%C A191432 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.
%C A191432 Examples:
%C A191432   (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191432   (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191432   (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191432 More recent examples of dispersions: A191426-A191455.
%C A191432 Conjecture: It appears this sequence is related to the even numbers with odd abundance A088827. Looking at the table format if the columns represent the powers of 2 (starting at 2^1) and the rows represent the squares of odd numbers, then taking the product of a term's row and column gives the n-th term in A088827. Example: A088827(67) = (7^2) * (2^6) = 3136. - _John Tyler Rascoe_, Jul 12 2022
%H A191432 Michel Marcus, <a href="/A191432/b191432.txt">Antidiagonals n = 1..100, flattened</a>
%e A191432 Northwest corner:
%e A191432    1    2    3    4    6    9
%e A191432    5    7   10   14   20   28
%e A191432    8   12   17   24   34   48
%e A191432   11   16   23   33   47   67
%e A191432   15   21   30   43   61   86
%t A191432 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191432 r = 40; r1 = 12;  (* r=# rows of T, r1=# rows to show *)
%t A191432 c = 40; c1 = 12;  (* c=# cols of T, c1=# cols to show *)
%t A191432 x = Sqrt[2];
%t A191432 f[n_] := Floor[n*x + 1/x] (* f(n) is complement of column 1 *)
%t A191432 mex[list_] :=  NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,
%t A191432   Length[Union[list]]]
%t A191432 rows = {NestList[f, 1, c]};
%t A191432 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191432 t[i_, j_] := rows[[i, j]];  TableForm[
%t A191432 Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191432 array *)
%t A191432 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191432 sequence *)
%t A191432 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%o A191432 (PARI) s(n) = my(x=quadgen(8)); floor(n*x+1/x);  \\ A001953
%o A191432 t(n) = floor((n+1/2)*(2+quadgen(8))); \\ A001954
%o A191432 T(n, k) = my(x = t(n-1)); for (i=2, k, x = s(x);); x; \\ _Michel Marcus_, Jul 13 2022
%Y A191432 Cf. A114537, A035513, A035506, A088827.
%Y A191432 Cf. A001953, A001954.
%K A191432 nonn,tabl
%O A191432 1,2
%A A191432 _Clark Kimberling_, Jun 03 2011