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A191433
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Dispersion of ([nx+n+1/2]), where x=(golden ratio) and [ ]=floor, by antidiagonals.
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1
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1, 3, 2, 8, 5, 4, 21, 13, 10, 6, 55, 34, 26, 16, 7, 144, 89, 68, 42, 18, 9, 377, 233, 178, 110, 47, 24, 11, 987, 610, 466, 288, 123, 63, 29, 12, 2584, 1597, 1220, 754, 322, 165, 76, 31, 14, 6765, 4181, 3194, 1974, 843, 432, 199, 81, 37, 15, 17711, 10946
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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...144
2....5....13...34...89...233
4....10...26...68...178..466
6....16...42...110..288..754
7....18...47...123..322..843
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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 = 1 + GoldenRatio;
f[n_] := Floor[n*x + 1/2] (* 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]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191433 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191433 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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