A070231 -id:A070231

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A003686

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

+10
15

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) 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

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

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

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

KEYWORD

nonn,easy,nice

AUTHOR

Vsevolod F. Lev, c. 1998

EXTENSIONS

Additional description from Andreas M. Hinz and Daniele Parisse

STATUS

approved

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).

+10
13

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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

LINKS

Table of n, a(n) for n=0..13.

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.

FORMULA

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

MATHEMATICA

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 *)

PROG

(PARI) {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])}

(PARI) {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)))}

CROSSREFS

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

See A236394 for the primes that are produced.

KEYWORD

nonn,easy

AUTHOR

Michael Somos, Oct 07 2001

STATUS

approved

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.

+10
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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

LINKS

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

FORMULA

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

MAPLE

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

MATHEMATICA

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 *)

PROG

(PARI) { 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

(Haskell)

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

(Sage)

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

(Magma) [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

CROSSREFS

The b(n) sequence is A003686.

See A094303 for another version.

Cf. A001697, A070231, A070233, A070234.

Cf. A001622 (golden ratio).

KEYWORD

nonn,easy

AUTHOR

Leroy Quet, Oct 31 2001

STATUS

approved

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).

+10
8

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Next term is too large to include.

LINKS

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

FORMULA

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

MATHEMATICA

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 *)

PROG

(PARI) 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

CROSSREFS

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

KEYWORD

easy,nonn

AUTHOR

Benoit Cloitre, May 08 2002

STATUS

approved

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).

+10
8

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

FORMULA

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

MATHEMATICA

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 *)

PROG

(PARI)_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

CROSSREFS

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

KEYWORD

easy,nonn

AUTHOR

Benoit Cloitre, May 08 2002

STATUS

approved

A094303

a(1) = 1, a(2) = 2, and a(n+1) = a(n) * sum of all previous terms up to a(n-1) for n >= 2.

+10
8

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

From Petros Hadjicostas, May 11 2020: (Start)

R. J. Mathar's conjecture is correct and this is identical to A064847 starting at n = 3. To see why this is the case, consider the sequences u(n) and v(n) defined by u(1) = v(1) = 1, and u(k+1) = u(k) + v(k), v(k+1) = u(k)*v(k) for k >= 1. Then u(n) = A003686(n) and v(n) = A064847(n) for n >= 1.

Then v(n) = u(n+1) - u(n), and thus Sum_{k=1..n-1} v(k) = u(n) - u(1) = u(n) - 1 for n >= 2. Then v(n-1) + ... + v(3) + (v(2) + 1) + v(1) = u(n) for n >= 3, and hence v(n)*(v(n-1) + ... + v(3) + (v(2) + 1) + v(1)) = u(n)*v(n) = v(n+1).

Since v(1) = 1 = a(1) and v(2) + 1 = 2 = a(2), the sequence (v(1), v(2) + 1, v(3), ..., v(n), ...) is identical to the current sequence. Hence, a(n) = v(n) = u(n+1) - u(n) = A003686(n+1) - A003686(n) for n >= 3. (End)

LINKS

Petros Hadjicostas, Table of n, a(n) for n = 1..17

FORMULA

Conjecture: a(n) = A003686(n+1) - A003686(n) for n >= 3. - R. J. Mathar, Apr 24 2007

MATHEMATICA

nxt[{t1_, t2_, a_}]:=Module[{c=t1*a}, {t1+t2, c, c}]; Join[{1}, NestList[nxt, {1, 2, 2}, 10][[All, 2]]] (* Harvey P. Dale, Aug 30 2020 *)

PROG

(PARI) lista(nn) = { my(va = vector(nn)); va[1] = 1; va[2] = 2; for(n=3, nn, va[n] = va[n-1]*sum(k=1, n-2, va[k]); ); va; } \\ Petros Hadjicostas, May 11 2020

CROSSREFS

See A064847 for another version.

Cf. A003686, A064183, A064526, A070231, A070233, A070234.

KEYWORD

nonn

AUTHOR

Amarnath Murthy, Apr 29 2004

EXTENSIONS

More terms from Gareth McCaughan, Jun 10 2004

STATUS

approved

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).

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..13.

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

FORMULA

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

MATHEMATICA

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 *)

PROG

(PARI) {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])}

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Michael Somos, Sep 20 2001

STATUS

approved

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