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A191436
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Dispersion of ([nx+n+x-1]), where x=(golden ratio) and [ ]=floor, by antidiagonals.
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2
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1, 4, 2, 12, 6, 3, 33, 17, 9, 5, 88, 46, 25, 14, 7, 232, 122, 67, 38, 19, 8, 609, 321, 177, 101, 51, 22, 10, 1596, 842, 465, 266, 135, 59, 27, 11, 4180, 2206, 1219, 698, 355, 156, 72, 30, 13, 10945, 5777, 3193, 1829, 931, 410, 190, 80, 35, 15, 28656, 15126
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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.....4....12...33...88
2.....6....17...46...122
3.....9....25...67...177
5.....14...38...101..266
7.....19...51...135..355
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MATHEMATICA
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(* Program generates the dispersion array T of increasing sequence f[n] *)
r = 40; r1 = 12; c = 40; c1 = 12; x = GoldenRatio;
f[n_] := Floor[n*x+n+x-1] (* 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}]] (* A191436 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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