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A248513 Rectangular array by antidiagonals: the dispersion of A181155 ("odious numbers").

%I A248513 #13 Oct 21 2024 01:20:28
%S A248513 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 A248513 39,22,13,65,225,177,105,77,43,26,16,129,449,353,209,153,85,51,32,18,
%U A248513 257,897,705,417,305,169,101,63,36,19,513,1793,1409,833,609,337
%N A248513 Rectangular array by antidiagonals: the dispersion of A181155 ("odious numbers").
%C A248513 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 A248513 The n-th term of column 1 is A001969(n) + 1, where A001969 are the "evil numbers".
%D A248513 Clark Kimberling, Fractal sequences and interspersions, Ars Combinatoria 45 (1997) 157-168.
%H A248513 Clark Kimberling, <a href="/A248513/b248513.txt">Antidiagonals n = 1..60, flattened</a>
%H A248513 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 A248513 Northwest corner:
%e A248513   1 ... 2 ... 3 ... 5 ... 9 .... 17 ... 33
%e A248513   4 ... 8 ... 15 .. 29 .. 57 ... 113 .. 225
%e A248513   6 ... 12 .. 23 .. 45 .. 89 ... 177 .. 353
%e A248513   7 ... 14 .. 27 .. 53 .. 105 .. 209 .. 417
%e A248513   10 .. 20 .. 39 .. 77 .. 153 .. 305 .. 609
%t A248513 r = 40; r1 = 10; (* r = # rows of T, r1 = # rows to show *);
%t A248513 c = 40; c1 = 12; (* c = # cols of T, c1 = # cols to show *);
%t A248513 x = GoldenRatio;
%t A248513 s[n_] := s[n] = If[n < 1, 0, 2 n - Mod[Total[IntegerDigits[n - 1, 2]], 2]];
%t A248513 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,   Length[Union[list]]]; rows = {NestList[s, 1, c]};
%t A248513 Do[rows = Append[rows, NestList[s, mex[Flatten[rows]], r]], {r}];
%t A248513 t[i_, j_] := rows[[i, j]]; TableForm[Table[t[i, j], {i, 1, r1}, {j, 1, c1}]]
%t A248513 u = Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A248513 *)
%t A248513 row[i_] := row[i] = Table[t[i, j], {j, 1, c}]
%t A248513 f[n_] := Select[Range[r], MemberQ[row[#], n] &]
%t A248513 v = Flatten[Table[f[n], {n, 1, 200}]]  (* A248514 *)
%Y A248513 Cf. A248514.
%K A248513 nonn,tabl,easy
%O A248513 1,2
%A A248513 _Clark Kimberling_, Oct 08 2014