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A191429
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Dispersion of ([n*sqrt(2)+2]), where [ ]=floor, by antidiagonals.
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2
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1, 3, 2, 6, 4, 5, 10, 7, 9, 8, 16, 11, 14, 13, 12, 24, 17, 21, 20, 18, 15, 35, 26, 31, 30, 27, 23, 19, 51, 38, 45, 44, 40, 34, 28, 22, 74, 55, 65, 64, 58, 50, 41, 33, 25, 106, 79, 93, 92, 84, 72, 59, 48, 37, 29, 151, 113, 133, 132, 120, 103, 85, 69, 54, 43, 32, 215, 161, 190, 188, 171, 147, 122, 99, 78, 62, 47, 36
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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...6...10..16
2...4...7...11..17
5...9...14..21..31
8...13..20..30..44
12..18..27..40..58
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MATHEMATICA
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(* Program generates the dispersion array T of increasing sequence f[n] *)
r = 40; r1 = 12; (* r=# rows of T to compute, r1=# rows to show *)
c = 40; c1 = 12; (* c=# cols to compute, c1=# cols to show *)
x = Sqrt[2];
f[n_] := Floor[n*x + 2] (* f(n) is 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]]; (* the array T *)
TableForm[
Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191429 array *)
Flatten[Table[
t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191429 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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