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A014675
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The infinite Fibonacci word (start with 1, apply 1->2, 2->21, take limit).
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36
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2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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COMMENTS
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The limiting mean and variance of the first n terms are both equal to the golden ratio (A001622). - Clark Kimberling, Mar 12 2014
Let F = A000045 (Fibonacci numbers). For n >= 3, the first F(n)-2 terms of A014675 form a palindrome; see A001911. If k is not one of the numbers F(n)-2, then the first k terms of A014675 do not form a palindrome. - Clark Kimberling, Jul 14 2014
First differences of A000201. - Tom Edgar, Apr 23 2015 [Editor's note: except for the offset: as for A022342, below. - M. F. Hasler, Oct 13 2017]
Also first differences of A022342 (which starts at offset 1): a(n)=A022342(n+2)-A022342(n+1), n >= 0. Equal to A001468 without its first term: a(n) = A001468(n+1), n >= 0. - M. F. Hasler, Oct 13 2017
The word is a concatenation of three runs: 1, 2, and 22. The limiting proportions of these are respectively 1/2, 1 - phi/2, and (phi - 1)/2, where phi = golden ratio. The mean runlength is (phi + 1)/2. -Clark Kimberling, Dec 26 2010
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REFERENCES
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D. Gault and M. Clint, "Curiouser and curiouser" said Alice. Further reflections on an interesting recursive function, Internat. J. Computer Math., 26 (1988), 35-43. See Table 2.
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7, p. 36.
G. Melancon, Factorizing infinite words using Maple, MapleTech journal, vol. 4, no. 1, 1997, pp. 34-42, esp. p. 36.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..10945 (20 iterations)
M. Bunder and K. Tognetti, On the self matching properties of [j tau], Discrete Math., 241 (2001), 139-151.
D. Gault & M. Clint, "Curiouser and curiouser said Alice. Further reflections on an interesting recursive function, Intern. J. Computer. Math., 26 (1988), 35-43. (Annotated scanned copy)
J. Grytczuk, Infinite semi-similar words, Discrete Math. 161 (1996), 133-141.
G. Melancon, Lyndon factorization of sturmian words, Discr. Math., 210 (2000), 137-149.
Index entries for sequences that are fixed points of mappings
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FORMULA
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Define strings S(0)=1, S(1)=2, S(n)=S(n-1).S(n-2) for n>=2. Sequence is S(infinity).
a(n) = floor((n+2)*phi) - floor((n+1)*phi) = A000201(n+2) - A000201(n+1), phi = (1 + sqrt(5))/2.
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MAPLE
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Digits := 50: t := evalf( (1+sqrt(5))/2); A014675 := n->floor((n+2)*t)-floor((n+1)*t);
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MATHEMATICA
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Nest[ Flatten[ # /. {1 -> 2, 2 -> {2, 1}}] &, {1}, 11] (* Robert G. Wilson v *)
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PROG
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(PARI) first(n)=my(v=[1], u); while(#v<n, u=List(); for(i=1, #v, listput(u, 2); if(v[i]==2, listput(u, 1))); v=Vec(u)); v[1..n] \\ Charles R Greathouse IV, Jun 21 2017
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CROSSREFS
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This is the 1, 2 version. The standard form is A003849. See also A005614.
Cf. A082389, A008351, A000045, A001622, A001911.
Equals A001468 except for initial term.
Differs from A025143 in many entries starting at entry 8.
First differences of A000201 and of A022342.
Sequence in context: A306260 A180227 A001468 * A308186 A107362 A166332
Adjacent sequences: A014672 A014673 A014674 * A014676 A014677 A014678
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Corrected by N. J. A. Sloane, Nov 07 2001
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STATUS
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approved
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