A001697 -id:A001697 - cp's OEIS frontend

A062962 Number of divisors of n-th term of sequence a(n+1) = a(n)*(a(0) + ... + a(n)) (A001697).

1, 1, 2, 4, 12, 40, 160, 792, 9408, 783360, 55987200, 35610624000, 269007298560000

Offset: 0

Jason Earls, Jul 22 2001

a[0] = 1; a[1] = 1; a[n_] := a[n] = a[n - 1]^2*(1 + 1/a[n - 2]); Table[DivisorSigma[0, a[n]], {n, 0, 10}]  (* Amiram Eldar, Feb 17 2019 after Jean-François Alcover at A001697 *)

a(n)=if(n<2,n >= 0,a(n-1)^2*(1+1/a(n-2)));
for(n=0,11,print1(numdiv(a(n)), ", "))

a(n) = A000005(A001697(n)). - Amiram Eldar, Feb 17 2019

a(11) from Amiram Eldar, Feb 17 2019

a(12) from Max Alekseyev, Feb 20 2024

A039941 Alternately add and multiply.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 4, 8, 12, 96, 108, 10368, 10476, 108615168, 108625644, 11798392572168192, 11798392680793836, 139202068568601556987554268864512, 139202068568601568785946949658348, 19377215893777651167043206536157390321290709180447278572301746176

Offset: 0

Walter Carlini

Reinhard Zumkeller, Table of n, a(n) for n = 0..27
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)

A001696(n)=A039941(2*n), A001697(n)=A039941(2*n+1).

Cf. A077753

a039941 n = a039941_list !! (n-1)
a039941_list = 0 : 1 : zipWith3 ($)
   (cycle [(+),(*)]) a039941_list (tail a039941_list)
-- Reinhard Zumkeller, May 07 2012

nxt[{n_,a_,b_}]:={n+1,b,If[EvenQ[n],a+b,a*b]}; Join[{0},Transpose[ NestList[ nxt,{0,0,1},20]][[3]]] (* Harvey P. Dale, Aug 23 2013 *)

a(n)=if(n<2,n>0, if(n%2,a(n-1)*a(n-2),a(n-1)+a(n-2)))

a(2n) = a(2n-1) + a(2n-2); a(2n+1) = a(2n-1)*a(2n); a(0) = 0; a(1) = 1

a(n) = {a(n-1) + a(n-2), n even, a(n-1)*a(n-2), n odd}; a(0)=0; a(1)=1.

Additional comments from Michael Somos, May 19 2000

One more term from Harvey P. Dale, Aug 23 2013

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

A001696 a(n) = a(n-1)*(1 + a(n-1) - a(n-2)), a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 2, 4, 12, 108, 10476, 108625644, 11798392680793836, 139202068568601568785946949658348, 19377215893777651167043206536157529523359277782016064519251404524

Offset: 0

N. J. A. Sloane

Also, numbers remaining after the following sieving process: In step 1, keep all numbers of the set N={0,1,2,...}. In step 2, keep only every second number after a(2)=2: N'={0,1,2,4,6,8,10,...}. In step 3, keep every 4th of the numbers following a(3)=4, N"={0,1,2,4,12,20,28,...}. In step 4, keep every 12th of the numbers beyond a(4)=12: {0,1,2,4,12,108,204,...}. In step 5, keep every 108th of the numbers beyond a(5)=108: {0,1,2,4,12,108,10476,...}, and so on. The next "gap" a(n+1)-a(n) is always a(n) times the former gap, i.e., a(n+1)-a(n) = a(n)*(a(n)-a(n-1)). - M. F. Hasler, Oct 28 2010

Number of plane trees where the root has fewer than n children and the i-th child of any node has fewer than i children. - David Eppstein, Dec 18 2021

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Cerkan, Table of n, a(n) for n = 0..13
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
Index entries for sequences of form a(n+1)=a(n)^2 + ...

a(n)=A039941(2*n); first difference sequence of this sequence is A001697. - Michael Somos, May 19 2000

a001696 n = a001696_list !! n
a001696_list = 0 : 1 : zipWith (-)
   (zipWith (+) a001696_list' $ map (^ 2) a001696_list')
   (zipWith (*) a001696_list a001696_list')
   where a001696_list' = tail a001696_list
-- Reinhard Zumkeller, Apr 29 2013

a[0] = 0; a[1] = 1; a[n_] := a[n] = a[n-1]*(1 + a[n-1] - a[n-2]); Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Jul 02 2013 *)

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

a(n) ~ c^(2^n), where c = 1.15552822483840350150537253088299651035583896919522349372370013726451673646... . - Vaclav Kotesovec, May 21 2015

cp's OEIS Frontend

A062962 Number of divisors of n-th term of sequence a(n+1) = a(n)*(a(0) + ... + a(n)) (A001697).

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A039941 Alternately add and multiply.

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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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A001696 a(n) = a(n-1)*(1 + a(n-1) - a(n-2)), a(0) = 0, a(1) = 1.

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