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A191431
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Dispersion of ([nx+x]), where x=sqrt(2) and [ ]=floor, by antidiagonals.
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1
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 13, 16, 18, 21, 22, 19, 17, 24, 26, 31, 32, 28, 25, 20, 35, 38, 45, 46, 41, 36, 29, 23, 50, 55, 65, 66, 59, 52, 42, 33, 27, 72, 79, 93, 94, 84, 74, 60, 48, 39, 30, 103, 113, 132, 134, 120, 106, 86, 69, 56, 43, 34, 147, 161, 188, 190, 171, 151, 123, 98, 80, 62, 49, 37
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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.....2....4....7...11...16
3.....5....8...12...18...26
6.....9...14...21...31...45
10...15...22...32...46...66
13...19...28...41...59...84
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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, r1=# rows to show *)
c = 40; c1 = 12; (* c=# cols of T, c1=# cols to show *)
x = Sqrt[2];
f[n_] := Floor[n*x + x] (* 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]];
TableForm[
Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
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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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