A094303 -id:A094303 - cp's OEIS frontend

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

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

Offset: 1

Vsevolod F. Lev, c. 1998

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

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

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

Cf. A001622, A001685, A064526, A064847, A070231, A070233, A070234, A094303.

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

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 *)

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])

Limit_{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/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

Additional description from Andreas M. Hinz and Daniele Parisse

A064526 Define a pair of sequences by p(0) = 0, q(0) = p(1) = q(1) = 1, q(n+1) = p(n)*q(n-1), p(n+1) = q(n+1) + q(n) for n > 0; then a(n) = p(n) and A064183(n) = q(n).

Original entry on oeis.org

0, 1, 2, 3, 5, 13, 49, 529, 21121, 10369921, 213952189441, 2214253468601687041, 473721461635593679669210030081, 1048939288228833101089604217183056027094304481281

Offset: 0

Michael Somos, Oct 07 2001

Every nonzero term is relatively prime to all others (which proves that there are infinitely many primes). See A236394 for the primes that appear.

R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - _N. J. A. Sloane_, Jun 13 2012
Michael Somos and R. Haas, A linked pair of sequences implies the primes are infinite, Amer. Math. Monthly, 110(6) (2003), 539-540.

Cf. A001685, A003686, A064183, A064847, A070231, A070233, A070234, A094303.

See A236394 for the primes that are produced.

Flatten[{0,1, RecurrenceTable[{a[n]==(a[n-1]^2 + a[n-2]^2 - a[n-1]*a[n-2] * (1+a[n-2]))/(1-a[n-2]), a[2]==2, a[3]==3},a,{n,2,15}]}] (* Vaclav Kotesovec, May 21 2015 *)

{a(n) = local(v); if( n<3, max(0, n), v = [1,1]; for( k=3, n, v = [v[2], v[1] * (v[1] + v[2])]); v[1] + v[2])}

{a(n) = if( n<4, max(0, n), (a(n-1)^2 + a(n-2)^2 - a(n-1) * a(n-2) * (1 + a(n-2))) / (1 - a(n-2)))}

a(n) = (a(n-1)^2 + a(n-2)^2 - a(n-1) * a(n-2) * (1 + a(n-2))) / (1 - a(n-2)) for n >= 2.

a(n) ~ c^(phi^n), where c = 1.2364241784241086061606568429916822975882631646194967549068405592472125928485... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 21 2015

A064847 Sequence a(n) such that there is a sequence b(n) with a(1) = b(1) = 1, a(n+1) = a(n) * b(n) and b(n+1) = a(n) + b(n) for n >= 1.

Original entry on oeis.org

1, 1, 2, 6, 30, 330, 13530, 5019630, 69777876630, 351229105131280530, 24509789089304573335878465330, 8608552999157278550998626549630446732052243030

Offset: 1

Leroy Quet, Oct 31 2001

Consider the mapping f(a/b) = (a + b)/(ab). 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 denominators. - Amarnath Murthy, Mar 24 2003

Harry J. Smith, Table of n, a(n) for n=1..18

The b(n) sequence is A003686.
See A094303 for another version.
Cf. A001697, A070231, A070233, A070234.
Cf. A001622 (golden ratio).

a064847 n = a064847_list !! (n-1)
a064847_list = 1 : f [1,1] where
   f xs'@(x:xs) = y : f (y : xs') where y = x * sum xs
-- Reinhard Zumkeller, Apr 29 2013

[n le 2 select 1 else Self(n-1)*(Self(n-1)/Self(n-2) + Self(n-2)): n in [1..14]]; // Vincenzo Librandi, Dec 17 2015

f:= proc(n) option remember; procname(n-1)*(procname(n-1)/procname(n-2) + procname(n-2)) end proc:
f(1):= 1: f(2):= 1:
map(f, [$1..16]); # Robert Israel, Jul 18 2016

RecurrenceTable[{a[n]==a[n-1]*(a[n-1]/a[n-2] + a[n-2]), a[0]==1, a[1]==1},a,{n,0,15}] (* Vaclav Kotesovec, May 21 2015 *)
Im[NestList[Re@#+(1+I Re@#)Im@#&, 1+I, 15]] (* Vladimir Reshetnikov, Jul 18 2016 *)

{ for (n=1, 18, if (n>2, a=a1*(a1/a2 + a2); a2=a1; a1=a, a=a1=a2=1); write("b064847.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 28 2009

def A064847():
    x, y = 1, 2
    yield x
    while True:
        yield x
        x, y = x * y, x + y
a = A064847()
[next(a) for i in range(12)]  # Peter Luschny, Dec 17 2015

a(n+2) = a(n+1)*(a(n+1)/a(n) + a(n)) for n >= 1 with a(1) = a(2) = 1.
Lim_{n -> infinity} a(n)/A003686(n)^phi = 1, where phi = (1 + sqrt(5))/2 is the golden ratio. - Benoit Cloitre, May 08 2002
Denominator of b(n), where b(n) = 1/numerator(b(n-1)) + 1/denominator(b(n-1)) for n >= 2 with b(1) = 1. Cf. A003686. - Vladeta Jovovic, Aug 15 2002
a(n) ~ c^(phi^n), where c = 1.70146471458872503754529013562504670973656402413202907200954401051557047249... and phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 21 2015

A070231 Let u(k), v(k), w(k) satisfy the recursions u(1) = v(1) = w(1) = 1, u(k+1) = u(k) + v(k) + w(k), v(k+1) = u(k)v(k) + v(k)w(k) + w(k)u(k), and w(k+1) = u(k)v(k)*w(k) for k >= 1; then a(n) = u(n).

Original entry on oeis.org

1, 3, 7, 31, 1279, 9202687, 3692849258577919, 98367959484921734629696721986125823, 3882894052327310957045599009145809243674851356642054390303168725061781159935999

Offset: 1

Benoit Cloitre, May 08 2002

Petros Hadjicostas, Table of n, a(n) for n = 1..12

Cf. A003686, A064183, A064526, A064847, A070233 (= w), A070234 (= v), A094303, A160389, A255249.

a[1] = 1; v[1] = 1; w[1] = 1; a[k_] := a[k] = a[k - 1] + v[k - 1] + w[k - 1]; v[k_] := v[k] = a[k - 1]*v[k - 1] + v[k - 1]*w[k - 1] + w[k - 1]*a[k - 1]; w[k_] := w[k] = a[k - 1]*v[k - 1]*w[k - 1]; Table[a[n], {n, 1, 9}] (* Vaclav Kotesovec, May 11 2020 *)

lista(nn) = {my(u = vector(nn)); my(v = vector(nn)); my(w = vector(nn)); u[1] = 1; v[1] = 1; w[1] = 1; for (n=2, nn, u[n] = u[n-1] + v[n-1] + w[n-1]; v[n] = u[n-1]*v[n-1] + v[n-1]*w[n-1] + w[n-1]*u[n-1]; w[n] = u[n-1]*v[n-1]*w[n-1];); u; } \\ Petros Hadjicostas, May 11 2020

Let C be the positive root of x^3 + x^2 - 2*x - 1 = 0; that is, C = 1.246979603717... = A255249. Then Lim_{n -> infinity} u(n)^(C+1)/w(n)= Lim_{n -> infinity} v(n)^C/w(n) = Lim_{n -> infinity} u(n)^B/v(n) = 1 with B = C + 1 - 1/(1 + C) = 1.8019377... = A160389. [corrected by Vaclav Kotesovec, May 11 2020]

a(n) ~ gu^((1 + C)^n), where C is defined above and gu = 1.131945853718244297... The relation between constants gu, gv (see A070234) and gw (see A070233) is gu^(1 + C) = gv^C = gw. - Vaclav Kotesovec, May 11 2020

A070233 Let u(k), v(k), w(k) satisfy the recursions u(1) = v(1) = w(1) = 1, u(k+1) = u(k) + v(k) + w(k), v(k+1) = u(k)v(k) + v(k)w(k) + w(k)u(k), and w(k+1) = u(k)v(k)*w(k) for k >= 1; then a(n) = w(n).

Original entry on oeis.org

1, 1, 9, 945, 8876385, 3689952451492545, 98367948795841301790914258556831105, 3882894052327309905582682317031276840071039865528864289025562807872336355445505

Offset: 1

Benoit Cloitre, May 08 2002

Next term is too large to include.

Petros Hadjicostas, Table of n, a(n) for n = 1..11

Cf. A003686, A064183, A064526, A064847, A070231 (= u), A070234 (= v), A094303, A160389, A255249.

u[1] = 1; v[1] = 1; a[1] = 1; u[k_] := u[k] = u[k - 1] + v[k - 1] + a[k - 1]; v[k_] := v[k] = u[k - 1]*v[k - 1] + v[k - 1]*a[k - 1] + a[k - 1]*u[k - 1]; a[k_] := a[k] = u[k - 1]*v[k - 1]*a[k - 1]; Table[a[n], {n, 1, 9}] (* Vaclav Kotesovec, May 11 2020 *)

lista(nn) = {my(u = vector(nn)); my(v = vector(nn)); my(w = vector(nn)); u[1] = 1; v[1] = 1; w[1] = 1; for (n=2, nn, u[n] = u[n-1] + v[n-1] + w[n-1]; v[n] = u[n-1]*v[n-1] + v[n-1]*w[n-1] + w[n-1]*u[n-1]; w[n] = u[n-1]*v[n-1]*w[n-1]; ); w; } \\ Petros Hadjicostas, May 11 2020

Let C be the positive root of x^3 + x^2 - 2*x - 1 = 0: that is, C = 1.246979603717... = A255249. Then Lim_{n -> infinity} u(n)^(C+1)/w(n)= Lim_{n -> infinity} v(n)^C/w(n) = Lim_{n -> infinity} u(n)^B/v(n) = 1 with B = C + 1 - 1/(1 + C) = 1.8019377... = A160389. [corrected by Vaclav Kotesovec, May 11 2020]

a(n) ~ gw^((C + 1)^n), where C is defined above and gw = 1.321128752475732548... The relation between constants gu (see A070231), gv (see A070234) and gw is gu^(1 + C) = gv^C = gw. - Vaclav Kotesovec, May 11 2020

A070234 Let u(k), v(k), w(k) satisfy the recursions u(1) = v(1) = w(1) = 1, u(k+1) = u(k) + v(k) + w(k), v(k+1) = u(k)v(k) + v(k)w(k) + w(k)u(k), and w(k+1) = u(k)v(k)*w(k); then a(n) = v(n).

Original entry on oeis.org

1, 3, 15, 303, 325023, 2896797882687, 10689080432835089614170716799, 1051462916692114532403603811392745230616355871287492722818364671

Offset: 1

Benoit Cloitre, May 08 2002

Petros Hadjicostas, Table of n, a(n) for n = 1..11

Cf. A003686, A064183, A064526, A064847, A070231 (= u), A070233 (= w), A094303, A160389, A255249.

u[1] = 1; a[1] = 1; w[1] = 1; u[k_] := u[k] = u[k - 1] + a[k - 1] + w[k - 1]; a[k_] := a[k] = u[k - 1]*a[k - 1] + a[k - 1]*w[k - 1] + w[k - 1]*u[k - 1]; w[k_] := w[k] = u[k - 1]*a[k - 1]*w[k - 1]; Table[a[n], {n, 1, 9}] (* Vaclav Kotesovec, May 11 2020 *)

lista(nn) = {my(u = vector(nn)); my(v = vector(nn)); my(w = vector(nn)); u[1] = 1; v[1] = 1; w[1] = 1; for (n=2, nn, u[n] = u[n-1] + v[n-1] + w[n-1]; v[n] = u[n-1]*v[n-1] + v[n-1]*w[n-1] + w[n-1]*u[n-1]; w[n] = u[n-1]*v[n-1]*w[n-1]; ); v; } \\ _Petros Hadjicostas, May 11 2020

Let C be the positive root of x^3 + x^2 - 2*x - 1 = 0; that is, C = 1.246979603717... = A255249. Then Lim_{n -> infinity} u(n)^(C+1)/w(n) = Lim_{n -> infinity} v(n)^C/w(n) = Lim_{n -> infinity} u(n)^B/v(n) = 1 with B = C + 1 - 1/(1 + C) = 1.8019377... = A160389. [corrected by Vaclav Kotesovec, May 11 2020]

a(n) ~ gv^((C + 1)^n), where C is defined above and gv = 1.250231610564761084... The relation between constants gu (see A070231), gv and gw (see A070233) is gu^(1 + C) = gv^C = gw. - Vaclav Kotesovec, May 11 2020

A064183 Define a pair of sequences by p(0) = 0, q(0) = p(1) = q(1) = 1, q(n+1) = p(n)*q(n-1), p(n+1) = q(n+1) + q(n) for n > 0; then a(n) = q(n) and A064526(n) = p(n).

Original entry on oeis.org

1, 1, 1, 2, 3, 10, 39, 490, 20631, 10349290, 213941840151, 2214253254659846890, 473721461633379426414550183191, 1048939288228833100615882755549676600679754298090

Offset: 0

Michael Somos, Sep 20 2001

Michael Somos and R. Haas, A linked pair of sequences implies the primes are infinite, Amer. Math. Monthly, 110(6) (2003), 539-540.

Cf. A001622, A003686, A064526, A064847, A070231, A070233, A070234, A094303, A236394.

Flatten[{1, RecurrenceTable[{a[n]==(a[n-1]+a[n-2])*a[n-2], a[1]==1, a[2]==1},a,{n,1,10}]}] (* Vaclav Kotesovec, May 21 2015 *)

{a(n) = local(v); if( n<3, n>=0, v = [1,1]; for( k=3, n, v = [v[2], v[1] * (v[1] + v[2])]); v[2])}

{a(n) = if( n<3, n>=0, (a(n-1) + a(n-2)) * a(n-2))}

a(n) = (a(n-1) + a(n-2))*a(n-2) for n >= 2.
Lim_{n -> infinity} a(n)/a(n-1)^phi = 1, where phi = A001622. - Gerald McGarvey, Aug 29 2004
a(n) ~ c^(phi^n), where c = 1.23642417842410860616065684299168229758826316461949675490684055924721259... and phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 21 2015

cp's OEIS Frontend

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

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A064526 Define a pair of sequences by p(0) = 0, q(0) = p(1) = q(1) = 1, q(n+1) = p(n)*q(n-1), p(n+1) = q(n+1) + q(n) for n > 0; then a(n) = p(n) and A064183(n) = q(n).

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A064847 Sequence a(n) such that there is a sequence b(n) with a(1) = b(1) = 1, a(n+1) = a(n) * b(n) and b(n+1) = a(n) + b(n) for n >= 1.

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A070231 Let u(k), v(k), w(k) satisfy the recursions u(1) = v(1) = w(1) = 1, u(k+1) = u(k) + v(k) + w(k), v(k+1) = u(k)*v(k) + v(k)*w(k) + w(k)*u(k), and w(k+1) = u(k)*v(k)*w(k) for k >= 1; then a(n) = u(n).

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A070233 Let u(k), v(k), w(k) satisfy the recursions u(1) = v(1) = w(1) = 1, u(k+1) = u(k) + v(k) + w(k), v(k+1) = u(k)*v(k) + v(k)*w(k) + w(k)*u(k), and w(k+1) = u(k)*v(k)*w(k) for k >= 1; then a(n) = w(n).

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A070234 Let u(k), v(k), w(k) satisfy the recursions u(1) = v(1) = w(1) = 1, u(k+1) = u(k) + v(k) + w(k), v(k+1) = u(k)*v(k) + v(k)*w(k) + w(k)*u(k), and w(k+1) = u(k)*v(k)*w(k); then a(n) = v(n).

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A064183 Define a pair of sequences by p(0) = 0, q(0) = p(1) = q(1) = 1, q(n+1) = p(n)*q(n-1), p(n+1) = q(n+1) + q(n) for n > 0; then a(n) = q(n) and A064526(n) = p(n).

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A070231 Let u(k), v(k), w(k) satisfy the recursions u(1) = v(1) = w(1) = 1, u(k+1) = u(k) + v(k) + w(k), v(k+1) = u(k)v(k) + v(k)w(k) + w(k)u(k), and w(k+1) = u(k)v(k)*w(k) for k >= 1; then a(n) = u(n).

A070233 Let u(k), v(k), w(k) satisfy the recursions u(1) = v(1) = w(1) = 1, u(k+1) = u(k) + v(k) + w(k), v(k+1) = u(k)v(k) + v(k)w(k) + w(k)u(k), and w(k+1) = u(k)v(k)*w(k) for k >= 1; then a(n) = w(n).

A070234 Let u(k), v(k), w(k) satisfy the recursions u(1) = v(1) = w(1) = 1, u(k+1) = u(k) + v(k) + w(k), v(k+1) = u(k)v(k) + v(k)w(k) + w(k)u(k), and w(k+1) = u(k)v(k)*w(k); then a(n) = v(n).