A013659 Initialized continued fraction for sqrt(n-th nonsquare) has period (1,a(n)).

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

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),...].

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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,...]

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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: A175938 A117016 A338255 * A016505 A304254 A010310

Adjacent sequences: A013656 A013657 A013658 * A013660 A013661 A013662

Keyword

nonn,more

Author

Clark Kimberling

Status

approved