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Search: id:A076840
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| A076840 |
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a(1) = a(2) = 1; a(n) = (a(n-1)+1)/a(n-2) (for n>2, n odd), (a(n-1)^2+1)/a(n-2) (for n>2, n even). |
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5
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| 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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Any sequence a(1),a(2),a(3),... defined by the recurrence a(n) = (a(n-1)+1)/a(n-2) (for n>2, n odd), (a(n-1)^2+1)/a(n-2) (for n>2, n even) has period 6. The theory of cluster algebras currently being developed by Fomin and Zelevinsky gives a context for these facts, but it doesn't really explain them in an elementary way. - James Propp, Nov 20, 2002
Terms of the simple continued fraction of 43/[3*sqrt(434)-37]. Decimal expansion of 16076/142857. [From Paolo P. Lava (ppl(AT)spl.at), Aug 05 2009]
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LINKS
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Sergey Fomin and Andrei Zelevinsky, Cluster algebras II: Finite type classification
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FORMULA
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a(n)=1/90*{29*(n mod 6)+29*[(n+1) mod 6]+44*[(n+2) mod 6]-31*[(n+3) mod 6]-[(n+4) mod 6]+14*[(n+5) mod 6]} with n>=0 - Paolo P. Lava (ppl(AT)spl.at), Nov 27 2006
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MAPLE
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a := 1; b := 1; f := proc(n) option remember; global a, b; if n=1 then RETURN(a); fi; if n=2 then RETURN(b); fi; if n mod 2 = 1 then RETURN((f(n-1)+1)/f(n-2)); fi; RETURN((f(n-1)^2+1)/f(n-2)); end;
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CROSSREFS
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Cf. A076839, A076841, A076844.
Sequence in context: A021399 A159897 A019709 this_sequence A078375 A065261 A130848
Adjacent sequences: A076837 A076838 A076839 this_sequence A076841 A076842 A076843
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Nov 21 2002
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