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A013659 Initialized continued fraction for sqrt(n-th nonsquare) has period (1,a(n)). 0
4, 2, 1, 8, 14, 4, 36, 18, 2, 9, 28, 6, 64, 32, 338, 16, 3, 392, 46, 8, 100, 50, 14, 25, 20, 3038 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The definition of initialized continued fraction depends on the usual recurrences for ordinary simple continued fractions:

...

(eq 1).... p(i)=a(i)*p(i-1)+p(i-2) for i>=2;

(eq 2)...  q(i)=a(i)*q(i-1)+q(i-2) for i>=2.

...

Suppose that p(0), q(0), p(1), q(1) are nonnegative integers such that {p(0), p(1)} is not {0} and {q(0), q(1)} is not {0}.  Suppose [b(0),b(1),b(2),...] is an ordinary simple continued fraction of a number B. The equations (1) and (2) with b(i) substituted for a(i) are used to define "initialized convergents" p(i)/q(i).  The limit of these exists and defines the "initialized continued fraction" of B, denoted by [(p(0),q(0),p(1),q(1)) ; b(0),b(1),b(2),...].

...

Examples:

sqrt(2)=[(1,1,6,4) ; 1,4,1,4,1,4,1,4,1,4,...]

sqrt(3)=[(1,1,4,2) ; 1,2,1,2,1,2,1,2,1,2,...]

sqrt(5)=[(1,1,3,1) ; 1,1,1,1,1,1,1,1,1,1,...]

sqrt(76)=[(2,2,3302,377) ; 1,338,1,338,1,338,...]

...

Formerly, an initialized continued fraction was here called a pseudo-continued fraction.  [From Clark Kimberling, Jun 23 2011]

REFERENCES

C. Kimberling, "Initialized continued fractions and Fibonacci numbers," in Proceedings of the Twelfth International Conference on Fibonacci Numbers and Their Applications, Congressus Numerantium 200 (2010) 269-284.

LINKS

Table of n, a(n) for n=1..26.

EXAMPLE

a(1)=4 because sqrt(2)=[(1,1,6,4) ; 1,4,1,4,1,4,...]

a(2)=2 because sqrt(3)=[(1,1,4,2) ; 1,2,1,2,1,2,...]

CROSSREFS

Sequence in context: A135294 A175938 A117016 * A016505 A010310 A021241

Adjacent sequences:  A013656 A013657 A013658 * A013660 A013661 A013662

KEYWORD

nonn,more

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified April 29 22:12 EDT 2017. Contains 285615 sequences.