%I #13 Oct 21 2024 01:20:28

%S 1,2,4,3,8,6,5,15,12,7,9,29,23,14,10,17,57,45,27,20,11,33,113,89,53,

%T 39,22,13,65,225,177,105,77,43,26,16,129,449,353,209,153,85,51,32,18,

%U 257,897,705,417,305,169,101,63,36,19,513,1793,1409,833,609,337

%N Rectangular array by antidiagonals: the dispersion of A181155 ("odious numbers").

%C 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, as in A248514.

%C The n-th term of column 1 is A001969(n) + 1, where A001969 are the "evil numbers".

%D Clark Kimberling, Fractal sequences and interspersions, Ars Combinatoria 45 (1997) 157-168.

%H Clark Kimberling, <a href="/A248513/b248513.txt">Antidiagonals n = 1..60, flattened</a>

%H Clark Kimberling, <a href="http://dx.doi.org/10.1090/S0002-9939-1993-1111434-0">Interspersions and dispersions</a>, Proceedings of the American Mathematical Society, 117 (1993) 313-321.

%e Northwest corner:

%e 1 ... 2 ... 3 ... 5 ... 9 .... 17 ... 33

%e 4 ... 8 ... 15 .. 29 .. 57 ... 113 .. 225

%e 6 ... 12 .. 23 .. 45 .. 89 ... 177 .. 353

%e 7 ... 14 .. 27 .. 53 .. 105 .. 209 .. 417

%e 10 .. 20 .. 39 .. 77 .. 153 .. 305 .. 609

%t r = 40; r1 = 10; (* r = # rows of T, r1 = # rows to show *);

%t c = 40; c1 = 12; (* c = # cols of T, c1 = # cols to show *);

%t x = GoldenRatio;

%t s[n_] := s[n] = If[n < 1, 0, 2 n - Mod[Total[IntegerDigits[n - 1, 2]], 2]];

%t mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]; rows = {NestList[s, 1, c]};

%t Do[rows = Append[rows, NestList[s, mex[Flatten[rows]], r]], {r}];

%t t[i_, j_] := rows[[i, j]]; TableForm[Table[t[i, j], {i, 1, r1}, {j, 1, c1}]]

%t u = Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A248513 *)

%t row[i_] := row[i] = Table[t[i, j], {j, 1, c}]

%t f[n_] := Select[Range[r], MemberQ[row[#], n] &]

%t v = Flatten[Table[f[n], {n, 1, 200}]] (* A248514 *)

%Y Cf. A248514.

%K nonn,tabl,easy

%O 1,2

%A _Clark Kimberling_, Oct 08 2014