A003285 Period of continued fraction for square root of n (or 0 if n is a square). (Formerly M0018)

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

Offset

1,3

Comments

Any string of five consecutive terms m^2 - 2 through m^2 + 2 for m > 2 in the sequence has the corresponding periods 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

Apparently the generating function of the sequence for the denominators of continued fraction convergents to sqrt(n) is always rational and of form p(x)/[1 - C*x^m + (-1)^m * x^(2m)], or equivalently, the denominators satisfy the linear recurrence b(n+2m) = C*b(n+m) - (-1)^m * b(n), where a(n) is equal to m for each nonsquare n, or 0. See A006702 for the conjecture regarding C. The same conjectures apply to the sequences of the numerators of continued fraction convergents to sqrt(n). - Ralf Stephan, Dec 12 2013

If a(n)=1, n is of form k^2+1 (A069987). See A013642 for a(n)=2, A013643 for a(n)=3, A013644 for a(n)=4, A010337 for a(n)=5, A020347 for a(n)=6, A010338 for a(n)=7, A020348 for a(n)=8, A010339 for a(n)=9, and furthermore A020349-A020439. - Ralf Stephan, Dec 12 2013

References

A. Brousseau, Number Theory Tables. Fibonacci Association, San Jose, CA, 1973, p. 197.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 5000 terms from T. D. Noe)

Marius Beceanu, Period of the Continued Fraction of sqrt(n), (Feb 05 2003)

Leon Bernstein, Fundamental units and cycles in the period of real quadratic number fields, I. Pacific J. Math 63 (1976): 37-61.

Ron Knott, All square-root continued fractions eventually repeat

R. Luczak, Patterns in the period lengths of simple periodic continued fractional representations of square roots of integers near perfect squares, QED: Chicago's Youth Math Research Symposium (April 2013).

R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018. - From N. J. A. Sloane, Jun 13 2012

Justin T. Miller, Families of Continued Fractions, translated (2000) from a document of Nikos Drakos, Computer Based Learning Unit, University of Leeds.

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.

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.

A. J. van der Poorten, Fractional Parts of the Period of the Continued Fraction Expansion of Quadratic Integers [Refined and revised text of a talk given at the 2nd Conference of the Canadian Number Theory Association, Vancouver, 1989]

A. J. van der Poorten, An introduction to continued fractions, Unpublished.

A. J. van der Poorten, An introduction to continued fractions, Unpublished [Cached copy]

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.

Maple

f:= n ->  if issqr(n) then 0
   else nops(numtheory:-cfrac(sqrt(n), 'periodic', 'quotients')[2]) fi:
map(f, [$1..100]); # Robert Israel, Sep 02 2015

Mathematica

a[n_] := ContinuedFraction[Sqrt[n]] // If[Length[ # ] == 1, 0, Length[Last[ # ]]]&
pcf[n_]:=Module[{s=Sqrt[n]}, If[IntegerQ[s], 0, Length[ContinuedFraction[s][[2]]]]]; Array[pcf, 110] (* Harvey P. Dale, Jul 15 2017 *)

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
(PARI) isok(n, p) = {localprec(p); my(cf = contfrac(sqrt(n))); setsearch(Set(cf), 2*cf[1]); }
a(n) = {if (issquare(n), 0, my(p=100); while (! isok(n, p), p+=100); localprec(p); my(cf = contfrac(sqrt(n))); for (k=2, #cf, if (cf[k] == 2*cf[1], return (k-1))); ); } \\ Michel Marcus, Jul 07 2021
(Python)
from sympy.ntheory.continued_fraction import continued_fraction_periodic
def a(n):
    cfp = continued_fraction_periodic(0, 1, d=n)
    return 0 if len(cfp) == 1 else len(cfp[1])
print([a(n) for n in range(1, 104)]) # Michael S. Branicky, Aug 22 2021

Crossrefs

Cf. A035015, A013943, A054269, A061490, A065938, A067280, A097853.

Sequence in context: A166692 A046766 A292147 * A059347 A071496 A071502

Adjacent sequences: A003282 A003283 A003284 * A003286 A003287 A003288

Keyword

nonn,nice

Author

N. J. A. Sloane

Status

approved