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A003686 Number of genealogical 1-2 rooted trees of height n. 14
1, 2, 3, 5, 11, 41, 371, 13901, 5033531, 69782910161, 351229174914190691, 24509789089655802510792656021, 8608552999157278575508415639286249242844899051 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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, 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, Mar 24 2003

An infinite coprime sequence defined by recursion. - Michael Somos, Mar 19 2004

REFERENCES

D. Parisse, The Tower of Hanoi and the Stern-Brocot Array, Thesis, Munich, 1997.

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

FORMULA

lim n -> infinity a(n)^phi/A064847(n)=1 where phi=(1+sqrt(5))/2 is the golden ratio. - Benoit Cloitre, 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.

a(n) ~ c^(phi^n), where c = 1.22508584062304325811405322247537613534139348463831009881946422737141574647... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 21 2015

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}] (* Harvey P. Dale, Jul 27 2011 *)

Re[NestList[Re@#+(1+I Re@#)Im@#&, 1+I, 15]] (* Vladimir Reshetnikov, Jul 18 2016 *)

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])

(MAGMA) I:=[1, 2]; [n le 2 select I[n] else Self(n-1)+Self(n-2)*(Self(n-1)-Self(n-2)): n in [1..14]]; // Vincenzo Librandi, Jul 19 2016

CROSSREFS

Cf. A001685, A064526, A064847.

Sequence in context: A188142 A276531 A276532 * A086506 A109462 A000905

Adjacent sequences:  A003683 A003684 A003685 * A003687 A003688 A003689

KEYWORD

nonn,easy,nice

AUTHOR

Vsevolod F. Lev, c. 1998

EXTENSIONS

Additional description from Andreas M. Hinz and Daniele Parisse

STATUS

approved

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Last modified October 23 20:17 EDT 2019. Contains 328373 sequences. (Running on oeis4.)