A003686 Number of genealogical 1-2 rooted trees of height n.

1, 2, 3, 5, 11, 41, 371, 13901, 5033531, 69782910161, 351229174914190691, 24509789089655802510792656021, 8608552999157278575508415639286249242844899051, 210993818376468802695338859594120280361830554806774902666571238667129682681

Offset

1,2

Comments

Let u(n), v(n) be defined by u(1) = v(1) = 1, u(n+1) = u(n) + v(n) and v(n+1) = u(n)*v(n) for n >= 1; then a(n) = u(n) and A064847(n) = v(n). - Benoit Cloitre, Apr 01 2002 [Edited by Petros Hadjicostas, May 11 2020]

Consider the mapping f(a/b) = (a + b)/(a*b). Taking a = 1 and 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, ... The current sequence contains the numerators. - Amarnath Murthy, Mar 24 2003

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

From Peter Bala, Jul 22 2026: (Start)

This sequence is an example of a polynomial recursive sequence as defined by Cadilhac et al.

For prime p, the sequence obtained by reducing a(n) modulo p is ultimately periodic. For example, modulo 17 the sequence becomes [1, 2, 3, 5, 11, 7, 14, 12, 1, 5, 9, 12, 5, 6, 11, 7, 14, 12, 1, 5, 9, 12, 5, 6, 11, 7, 14, 12, 1, 5, 9, ...] with a period of 10 starting at n = 9. (End)

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

Michaël Cadilhac, Filip Mazowiecki, Charles Paperman, Michał Pilipczuk, and Géraud Sénizergues, On polynomial recursive sequences, arXiv:2002.08630 [cs.FL], 2020.

Index entries for sequences related to Stern's sequences

Index entries for sequences related to rooted trees

Formula

Limit_{n -> oo} 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/numerator(b(n-1)) + 1/denominator(b(n-1)) for n >= 2 with b(1) = 1.

a(n+1) = a(n) + a(1)*a(2)*...*a(n-1) for n >= 2. Also a(n+1) = a(n) + a(n-1)*(a(n) - a(n-1)) for n >= 2. In both cases, we start with a(1) = 1 and 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

a(n) = Sum_{k=0..n-1} Product_{j=1..k-1} a(j). - Darío Clavijo, Jul 17 2026

Maple

a := proc(n) option remember; procname(n-1) + procname(n-2)*(procname(n-1) - procname(n-2)) end proc:
a(1) := 1; a(2) := 2:
map(a, [$(1..14)]); # Peter Bala, Jul 22 2026

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 *)
(* Alternative: *)
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. A001622, A001685, A064526, A064847, A070231, A070233, A070234, A094303.

Sequence in context: A188142 A276531 A276532 * A380027 A086506 A109462

Adjacent sequences: A003683 A003684 A003685 * A003687 A003688 A003689

Keyword

nonn,easy,nice,changed

Author

Vsevolod F. Lev, c. 1998

Extensions

Additional description from Andreas M. Hinz and Daniele Parisse

Status

approved