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A003686
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Number of genealogical 1-2 rooted trees of height n.
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13
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1, 2, 3, 5, 11, 41, 371, 13901, 5033531, 69782910161, 351229174914190691, 24509789089655802510792656021, 8608552999157278575508415639286249242844899051
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Let u(n), v(n) be defined by u(1)=v(1)=1, u(n+1)=u(n)+v(n), v(n+1)=u(n)*v(n); then a(n)=u(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 01 2002
Consider the mapping f(a/b) = (a + b)/(a*b). Taking a = 1 b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 1/1, 2/1, 3/2, 5/6, 11/30, ... Sequence contains the numerators. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 24 2003
An infinite coprime sequence defined by recursion. - Michael Somos Mar 19 2004
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REFERENCES
| D. Parisse, The Tower of Hanoi and the Stern-Brocot Array, Thesis, Munich, 1997.
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LINKS
| Franklin T. Adams-Watters, Table of n, a(n) for n = 1..19
Index entries for sequences related to Stern's sequences
Index entries for sequences related to rooted trees
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FORMULA
| lim n -> infinity a(n)^PHI/A064847(n)=1 where PHI=(1+sqrt(5))/2 is the golden ratio. - Benoit Cloitre (benoit7848c(AT)orange.fr), May 08 2002
Numerator of b(n) where b(n) = 1/numer(b(n-1)) + 1/denom(b(n-1)), b(1)=1.
a(n+1)=a(n)+a(1)*a(2)*...*a(n-1). Also a(n+1)=a(n)+a(n-1)*(a(n)-a(n-1)), a(1)=1, a(2)=2.
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MATHEMATICA
| RecurrenceTable[{a[1]==1, a[2]==2, a[n]==a[n-1]+a[n-2](a[n-1]-a[n-2])}, a[n], {n, 15}] (* From Harvey P. Dale, Jul 27 2011 *)
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PROG
| (PARI) a(n)=local(an); if(n<1, 0, an=vector(max(2, n)); an[1]=1; an[2]=2; for(k=3, n, an[k]=an[k-1]-an[k-2]^2+an[k-1]*an[k-2]); an[n])
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CROSSREFS
| Cf. A001685, A064526, A064847.
Sequence in context: A127181 A113734 A188142 * A086506 A109462 A000905
Adjacent sequences: A003683 A003684 A003685 * A003687 A003688 A003689
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Seva Lev (seva(AT)math.uga.edu)
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EXTENSIONS
| Additional description from Andreas M. Hinz and Daniele Parisse (hinz(AT)appl-math.tu-muenchen.de).
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