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,...]
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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.
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
KEYWORD
nonn,more
AUTHOR
STATUS
approved