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A191426
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Dispersion of (3+[nr]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.
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91
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1, 4, 2, 9, 6, 3, 17, 12, 7, 5, 30, 22, 14, 11, 8, 51, 38, 25, 20, 15, 10, 85, 64, 43, 35, 27, 19, 13, 140, 106, 72, 59, 46, 33, 24, 16, 229, 174, 119, 98, 77, 56, 41, 28, 18, 373, 284, 195, 161, 127, 93, 69, 48, 32, 21, 606, 462, 318, 263, 208, 153, 114, 80, 54, 36, 23, 983, 750, 517, 428, 339, 250, 187, 132, 90, 61, 40, 26
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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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REFERENCES
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Clark Kimberling, Fractal sequences and interspersions, Ars Combinatoria 45 (1997) 157-168.
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LINKS
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Mohammad K. Azarian, Problem 123, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3, Fall 1998, p. 176. Solution published in Vol. 12, No. 1, Winter 2000, pp. 61-62.
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EXAMPLE
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Northwest corner:
1...4...9...17..30
2...6...12..22..38
3...7...14..25..43
5...11..20..35..59
8...15..27..46..77
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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 = GoldenRatio; f[n_] := Floor[n*x + 3]
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}]] (* A191426 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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