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A191537
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Dispersion of (4n-floor(n*sqrt(2))), by antidiagonals.
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1
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1, 3, 2, 8, 6, 4, 21, 16, 11, 5, 55, 42, 29, 13, 7, 143, 109, 75, 34, 19, 9, 370, 282, 194, 88, 50, 24, 10, 957, 730, 502, 228, 130, 63, 26, 12, 2475, 1888, 1299, 590, 337, 163, 68, 32, 14, 6400, 4882, 3359, 1526, 872, 422, 176, 83, 37, 15, 16550, 12624
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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. Examples:
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LINKS
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EXAMPLE
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Northwest corner:
1, 3, 8, 21, 55, ...
2, 6, 16, 42, 109, ...
4, 11, 29, 75, 194, ...
5, 13, 34, 88, 228, ...
7, 19, 50, 130, 337, ...
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MATHEMATICA
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(* Program generates the dispersion array T of the increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; f[n_] :=4n-Floor[n*Sqrt[2]] (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, r1}, {j, 1, c1}]] (* A191537 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191537 sequence *)
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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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