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A020903
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Lim f(f(...f(n))) where f is the fractal sequence given by f(n)=A002260(n+1).
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5
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1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Suppose that f(1), f(2), f(3),... is a fractal sequence (a sequence which contains itself as a proper subsequence, such as 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, ...; if the first occurrence of each n is deleted, the remaining sequence is identical to the original; see the Wikipedia article for details). Then for each n>=1, the limit L(n) of composites f(f(f...f(n)...)) exists and is one of the numbers in the set {k : f(k)=k}. If f(2)>2, then L(n)=1 for all n; if f(2)=2 and f(3)>3, then L(n) is 1 or 2 for all n, etc. Examples: A020903, A191770, A191774.
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REFERENCES
| C. Kimberling, "Numeration systems and fractal sequences," Acta Arithmetica 73 (1995) 103-117.
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LINKS
| C. Kimberling, Fractal sequences
Wikipedia, Fractal sequence
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EXAMPLE
| f=(1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,2,3,4,5,6,...); write
n->n1->n2-> to mean n1=f(n), n2=f(n1),... Then
1->1, 2->2, 3->1, 4->2, 5->3->1, 6->1, 7->2, ...
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MATHEMATICA
| m[n_] := Floor[(-1 + Sqrt[8 n - 7])/2];
b[n_] := n - m[n] (m[n] + 1)/2; f[n_] := b[n + 1];
Table[m[n], {n, 1, 100}] (*A003056*)
Table[f[n], {n, 1, 100}] (*A002260(n+1)*)
h[n_] := Nest[f, n, 40]
t = Table[h[n], {n, 1, 300}] (* A020903 *)
Flatten[Position[t, 1]] (* A191777 *)
Flatten[Position[t, 2]] (* A020904 *)
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CROSSREFS
| Cf. A020904, A191777, A191770, A191774.
Sequence in context: A175922 A006337 A006338 * A133083 A083921 A105496
Adjacent sequences: A020900 A020901 A020902 * A020904 A020905 A020906
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KEYWORD
| nonn
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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