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A003285
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Period of continued fraction for square root of n (or 0 if n is a square).
(Formerly M0018)
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27
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0, 1, 2, 0, 1, 2, 4, 2, 0, 1, 2, 2, 5, 4, 2, 0, 1, 2, 6, 2, 6, 6, 4, 2, 0, 1, 2, 4, 5, 2, 8, 4, 4, 4, 2, 0, 1, 2, 2, 2, 3, 2, 10, 8, 6, 12, 4, 2, 0, 1, 2, 6, 5, 6, 4, 2, 6, 7, 6, 4, 11, 4, 2, 0, 1, 2, 10, 2, 8, 6, 8, 2, 7, 5, 4, 12, 6, 4, 4, 2, 0, 1, 2, 2, 5, 10, 2, 6, 5, 2, 8, 8, 10, 16, 4, 4, 11, 4, 2, 0, 1, 2, 12
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Any string of five consecutive terms m^2 - 2 through m^2 + 2 for m>2 in the sequence has the corresponding period lengths 4,2,0,1,2. - Lekraj Beedassy, Jul 17 2001
For m>1, a(m^2+m) = 2 and the continued fraction is m,2,2*m,2,2*m,2,2*m,... - Arran Fernandez, Aug 14 2011
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REFERENCES
| A. Brousseau, Number Theory Tables. Fibonacci Association, San Jose, CA, 1973, p. 197.
C. D. Patterson and H. C. Williams, "Some Periodic Continued Fractions with Long Periods," Mathematics of Computation, Vol. 44 (1985), No. 170, pp. 523-532.
A. M. Rockett and P. Szuesz, On the lengths of the periods of the continued fractions of square-roots of integers, Forum Mathematicum, 2 (1990), 119-123.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. G. Stanton, C. Sudler, Jr. and H. C. Williams, "An Upper Bound for the Period of the Simple Continued Fraction for Sqrt(D)," Pacific Journal of Math., Vol. 67 (1976), No. 2, pp. 525-536.
Hanna Uscka-Wehlou, Continued Fractions and Digital Lines with Irrational Slopes, in Discrete Geometry for Computer Imagery, Lecture Notes in Computer Science, Volume 4992/2008, Springer-Verlag. [From N. J. A. Sloane, Jul 10 2009]
H. C. Williams, "A Numerical Investigation Into the Length of the Period of the Continued Fraction Expansion of Sqrt(D)," Mathematics of Computation, Vol. 36 (1981), No. 154, pp. 593-601.
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LINKS
| T. D. Noe, Table of n, a(n) for n = 1..5000
M. Beceanu, Period of the Continued Fraction of sqrt(n)
Ron Knott, All square-root continued fractions eventually repeat
Justin T. Miller, Families of Continued Fractions
A. J. van der Poorten, An introduction to continued fractions
A. J. van der Poorten, Fractional parts of the period of the ...
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MATHEMATICA
| a[n_] := ContinuedFraction[Sqrt[n]] // If[Length[ # ] == 1, 0, Length[Last[ # ]]]&
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PROG
| (PARI) a(n)=if(issquare(n), return(0)); my(s=sqrt(n), x=s, f=floor(s), P=[0], Q=[1], k); while(1, k=#P; P=concat(P, f*Q[k]-P[k]); Q=concat(Q, (n-P[k+1]^2)/Q[k]); k++; for(i=1, k-1, if(P[i]==P[k]&&Q[i]==Q[k], return(k-i))); x=(P[k]+s)/Q[k]; f=floor(x)) \\ Charles R Greathouse IV, Jul 31 2011
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CROSSREFS
| Cf. A035015, A013943, A054269, A061490, A065938, A067280, A097853.
Sequence in context: A144172 A166692 A046766 * A059347 A071496 A071502
Adjacent sequences: A003282 A003283 A003284 * A003286 A003287 A003288
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KEYWORD
| nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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