This site is supported by donations to The OEIS Foundation.

"Email this user" was broken Aug 14 to 9am Aug 16. If you sent someone a message in this period, please send it again.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.