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A248513
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Rectangular array by antidiagonals: the dispersion of A181155 ("odious numbers").
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4
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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, 39, 22, 13, 65, 225, 177, 105, 77, 43, 26, 16, 129, 449, 353, 209, 153, 85, 51, 32, 18, 257, 897, 705, 417, 305, 169, 101, 63, 36, 19, 513, 1793, 1409, 833, 609, 337
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OFFSET
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1,2
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COMMENTS
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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.
The n-th term of column 1 is A001969(n) + 1, where A001969 are the "evil numbers".
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REFERENCES
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Clark Kimberling, Fractal sequences and interspersions, Ars Combinatoria 45 (1997) 157-168.
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LINKS
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EXAMPLE
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Northwest corner:
1 ... 2 ... 3 ... 5 ... 9 .... 17 ... 33
4 ... 8 ... 15 .. 29 .. 57 ... 113 .. 225
6 ... 12 .. 23 .. 45 .. 89 ... 177 .. 353
7 ... 14 .. 27 .. 53 .. 105 .. 209 .. 417
10 .. 20 .. 39 .. 77 .. 153 .. 305 .. 609
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MATHEMATICA
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r = 40; r1 = 10; (* r = # rows of T, r1 = # rows to show *);
c = 40; c1 = 12; (* c = # cols of T, c1 = # cols to show *);
x = GoldenRatio;
s[n_] := s[n] = If[n < 1, 0, 2 n - Mod[Total[IntegerDigits[n - 1, 2]], 2]];
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]; rows = {NestList[s, 1, c]};
Do[rows = Append[rows, NestList[s, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]]; TableForm[Table[t[i, j], {i, 1, r1}, {j, 1, c1}]]
u = Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A248513 *)
row[i_] := row[i] = Table[t[i, j], {j, 1, c}]
f[n_] := Select[Range[r], MemberQ[row[#], n] &]
v = Flatten[Table[f[n], {n, 1, 200}]] (* A248514 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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