A007660 -id:A007660 - cp's OEIS frontend

A121253 a(n) = a(n-1)*a(n-3)+1 with a(0)=a(1)=a(2)=0.

0, 0, 0, 1, 1, 1, 2, 3, 4, 9, 28, 113, 1018, 28505, 3221066, 3279045189, 93469183112446, 301070407771273987437, 987223472152664180906141290594, 92274971491542102812339702600558728264132925

Offset: 0

Jonathan Vos Post, Aug 22 2006

Analog of A007660 a(n)=a(n-1)*a(n-2)+1. What is the equivalent continued fraction and asymptotic representation?

Vincenzo Librandi, Table of n, a(n) for n = 0..26 (shortened by _N. J. A. Sloane_, Jan 13 2019)

Cf. A007660.

I:=[0,0,0]; [n le 3 select I[n] else Self(n-1)*Self(n-3)+1: n in [1..20]]; // Vincenzo Librandi, Nov 14 2011

RecurrenceTable[{a[0]==a[1]==a[2]==0,a[n]==a[n-1]a[n-3]+1},a,{n,20}] (* Harvey P. Dale, Dec 30 2011 *)

More terms from Vincenzo Librandi, Nov 14 2011

A121256 a(n) = a(n-1)*a(n-3) - 1, starting with a(0)=a(1)=a(2)=2.

Original entry on oeis.org

2, 2, 2, 3, 5, 9, 26, 129, 1160, 30159, 3890510, 4512991599, 136107313634240, 529526864767147062399, 2389750292138943783804215786000, 325262492519671886357848434144628838112639999

Offset: 0

Jonathan Vos Post, Aug 22 2006

Analog of A055937 a(n) = a(n-1)*a(n-2) - 1. What is the equivalent continued fraction and asymptotic representation, by analogy to A007660 a(n) = a(n-1)*a(n-2) + 1 ?

Vincenzo Librandi, Table of n, a(n) for n = 0..22 (shortened by _N. J. A. Sloane_, Jan 13 2019)

Cf. A007660, A055937.

I:=[2,2,2]; [n le 3 select I[n] else Self(n-1)*Self(n-3)-1: n in [1..20]]; // Vincenzo Librandi, Nov 14 2011

RecurrenceTable[{a[0]==a[1]==a[2]==2,a[n]==a[n-1]a[n-3]-1},a,{n,20}] (* Harvey P. Dale, Sep 02 2016 *)

a(n) = if(n<3, 2, a(n-1)*a(n-3) - 1);
vector(20, n, n--; a(n)) \\ G. C. Greubel, Jun 07 2019

def a(n):
    if (n==0 or n==1 or n==2): return 2
    else: return a(n-1)*a(n-3) -1
[a(n) for n in (0..20)] # G. C. Greubel, Jun 07 2019

Corrected and extended by Vincenzo Librandi, Nov 14 2011

A262714 a(n) = a(n-1)*a(n-2) + 1, with a(0) = a(1) = 2.

Original entry on oeis.org

2, 2, 5, 11, 56, 617, 34553, 21319202, 736642386707, 15704627843968647815, 11568694537326272321321120595206, 181682042349262169758803442669575561298555791374891, 2101824050856189730969091901210449068013789839106586804501928241686514359003372547

Offset: 0

Vincenzo Librandi, Sep 30 2015

G. C. Greubel, Table of n, a(n) for n = 0..17

Cf. A007660, A055937.

[n le 2 select 2 else Self(n-1)*Self(n-2)+1: n in [1..20]];

RecurrenceTable[{a[0]==a[1]==2, a[n]==a[n-1]*a[n-2] +1}, a, {n, 0, 20}]

a(n) = if(n<2, 2, 1 + a(n-1)*a(n-2))
vector(20, n, a(n-1)) \\ Altug Alkan, Sep 30 2015

{a(n) = if( n<2, 2 * (n>=0), self()(n-1) * self()(n-2) + 1)}; /* Michael Somos, Oct 02 2015 */

def a(n):
    if (n==0 or n==1): return 2
    else: return a(n-1)*a(n-2) +1
[a(n) for n in (0..20)] # G. C. Greubel, Jun 07 2019

A356891 a(n) = a(n-1) * a(n-2) + 1 if n is even, otherwise a(n) = a(n-3) + 1, with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 2, 2, 5, 3, 16, 6, 97, 17, 1650, 98, 161701, 1651, 266968352, 161702, 43169316455105, 266968353, 11524841314155180292066, 43169316455106, 497519521785644682185076928856988997, 11524841314155180292067

Offset: 0

J. Conrad, Sep 02 2022

nonn

			For n=2, a(2) = a(0) * a(1) + 1 = 2.
For n=3, a(3) = a(0) + 1 = 2.
For n=4, a(4) = a(3) * a(2) + 1 = 5.

J. Conrad, Table of n, a(n) for n = 0..34

Cf. A007660.

a[n_] := a[n] = If[EvenQ[n], a[n - 1]*a[n - 2], a[n - 3]] + 1; a[0] = a[1] = 1; Array[a, 22, 0] (* Amiram Eldar, Sep 10 2022 *)

def A356891(length):
    output = [1] * length
    for n in range(2, length):
        output[n] += output[n-3] if n % 2 else output[n-1] * output[n-2]
    return output

A364663 a(n+1) = a(|n-a(n)*a(n-1)|)+1; a(0) = 0.

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 2, 1, 4, 3, 2, 3, 4, 1, 4, 3, 2, 3, 4, 3, 2, 5, 4, 3, 4, 5, 4, 3, 4, 3, 4, 5, 4, 5, 2, 5, 6, 3, 4, 5, 4, 3, 4, 5, 4, 5, 6, 3, 4, 7, 6, 5, 6, 5, 4, 3, 6, 5, 4, 5, 6, 5, 6, 5, 6, 3, 4, 5, 4, 5, 8, 5, 6, 5, 6, 5, 6, 7, 6, 7, 4, 7, 6, 5, 6, 5, 4, 5, 6, 5, 6, 7, 8, 7, 4, 5, 6

Offset: 0

Rok Cestnik, Aug 01 2023

nonn

a(-1) can be set to any finite value and it does not affect the sequence.

			a(1) = a(|0-a(0)*____|)+1 = a(0)+1 = 1.
a(2) = a(|1-a(1)*a(0)|)+1 = a(1)+1 = 2.
a(3) = a(|2-a(2)*a(1)|)+1 = a(0)+1 = 1.

Cf. A364197, A340224, A339929.

Cf. A005185, A004001, A003056, A340134, A007660.

N=100; a=vector(N); a[2]=1; for(n=1,N-2,a[n+2]=a[1+abs(n-a[n]*a[n+1])]+1);

a=[0];
for n in range(100):
    a.append(a[abs(n-a[n]*a[n-1])]+1)

a(n) ~ (3*n)^(1/3) (conjectured).

cp's OEIS Frontend

A121253 a(n) = a(n-1)*a(n-3)+1 with a(0)=a(1)=a(2)=0.

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A121256 a(n) = a(n-1)*a(n-3) - 1, starting with a(0)=a(1)=a(2)=2.

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A262714 a(n) = a(n-1)*a(n-2) + 1, with a(0) = a(1) = 2.

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A356891 a(n) = a(n-1) * a(n-2) + 1 if n is even, otherwise a(n) = a(n-3) + 1, with a(0) = a(1) = 1.

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A364663 a(n+1) = a(|n-a(n)*a(n-1)|)+1; a(0) = 0.

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