A007660 -id:A007660 - cp's OEIS frontend

A258113 Decimal expansion of a constant related to A007660.

1, 1, 1, 3, 0, 5, 7, 9, 7, 5, 9, 0, 2, 9, 3, 1, 9, 3, 2, 8, 5, 3, 5, 9, 7, 7, 0, 7, 1, 6, 7, 5, 8, 4, 9, 1, 9, 0, 6, 6, 0, 0, 1, 8, 1, 5, 1, 0, 1, 8, 6, 5, 2, 7, 2, 0, 1, 4, 3, 7, 9, 7, 2, 4, 2, 0, 6, 9, 2, 7, 7, 1, 7, 2, 9, 7, 9, 8, 8, 2, 5, 9, 3, 8, 1, 6, 0, 9, 3, 6, 1, 4, 5, 4, 4, 5, 9, 4, 3, 5, 2, 2, 3, 4, 5

Offset: 1

Vaclav Kotesovec, May 20 2015

			1.1130579759029319328535977071675849190660018151018652720143797242069...

Cf. A007660, A076949, A250309, A258112.

A007660 = RecurrenceTable[{a[1]==0, a[2]==N[1,200], a[n]==a[n-1]*a[n-2]+1},a[n],{n,1,30}]; Do[Print[N[Exp[c2]/.Solve[Table[Log[A007660[[n]]]==c1*((1-Sqrt[5])/2)^n + c2*((1+Sqrt[5])/2)^n, {n,k,k+1}]], 120][[1]]],{k, Length[A007660]-2, Length[A007660]-1}];

Equals limit n->infinity (A007660(n))^((2/(1+sqrt(5)))^n).

A333974 Eventual period of A007660 modulo n.

Original entry on oeis.org

1, 3, 4, 3, 7, 12, 5, 3, 8, 21, 6, 12, 15, 15, 28, 3, 31, 24, 23, 21, 20, 6, 13, 12, 28, 15, 24, 15, 46, 84, 7, 3, 12, 93, 35, 24, 20, 69, 60, 21, 25, 60, 44, 6, 56, 39, 14, 12, 30, 84, 124, 15, 84, 24, 42, 15, 92, 138, 11, 84, 117, 21, 40, 3, 105, 12, 121, 93

Offset: 1

Isaac Kaufmann, Sep 03 2020

Multiplicative: If n = p^x * q^y * ... for distinct primes p,q,... then a(n) = a(p^x) * a(q^y) * ...

			a(1) is trivially 1.
For n=2 the sequence is 0, {0,1,1}, {0,1,1}, hence a(2) = 3.
For n=3 the sequence is 0, {0,1,1,2}, {0,1,1,2}, hence a(3) = 4.
For n=4 the sequence is 0,0,1,1, {2,3,3}, {2,3,3}, hence a(4) = 3.
For n=5 the sequence is 0, {0,1,1,2,3,2,2}, {0,1,1,2,3,2,2}, hence a(5) = 7.

David A. Corneth, PARI program.

See Corneth link \\ David A. Corneth, Sep 04 2020

a(n) | a(k*n), k=2,3,...

More terms from Jinyuan Wang, Sep 04 2020

A055937 a(n) = a(n-1) * a(n-2) - 1.

Original entry on oeis.org

2, 2, 3, 5, 14, 69, 965, 66584, 64253559, 4278258972455, 274893365303916717344, 1176065006579831683931038116759519, 323292467474902888912311701915651931811127805144397535

Offset: 1

Robert G. Wilson v, Jul 05 2000

Vincenzo Librandi, Table of n, a(n) for n = 1..19

A152446 uses largest prime < a(n-1)*a(n-2) instead of subtracting 1. - Joshua D. Olson, Sep 29 2015

Cf. A007660, A121258.

I:=[2,2]; [n le 2 select I[n] else Self(n-1)*Self(n-2)-1: n in [1..15]]; // Vincenzo Librandi, Sep 30 2015

RecurrenceTable[{a[1]==a[2]==2, a[n]==a[n-1]*a[n-2] -1}, a, {n, 15}] (* Vincenzo Librandi, Sep 30 2015 *)
nxt[{a_,b_}]:={b,a*b-1}; NestList[nxt,{2,2},15][[All,1]] (* Harvey P. Dale, Dec 05 2020 *)

a(n) = if(n<3, 2, a(n-1)*a(n-2)-1);
vector(15, n, a(n)) \\ Altug Alkan, Sep 30 2015

@cached_function
def a(n):
    if n == 1 or n == 2:
        return 2
    else:
        return a(n - 1) * a(n - 2) - 1
[a(n) for n in range(1, 16)]  # G. C. Greubel, Jun 07 2019

a(n) ~ c^(phi^n), where c = 1.26679081808631741720378389711... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 15 2019

A250309 a(n) = a(n-1)*(1 + a(n-1)/a(n-3)), with a(0) = a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 1, 2, 6, 42, 924, 143220, 488523420, 258285263294520, 465795819523189050504840, 444125576385425970712647062585372630520, 763680920404535561780141108036287312478667174369871222219397040

Offset: 0

Michael Somos, Jan 16 2015

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..17

Cf. A007660, A258113.

I:=[1,1,1]; [n le 3 select I[n] else Self(n-1)*(1 + Self(n-1)/Self(n-3)): n in [1..15]]; // G. C. Greubel, Aug 03 2018

RecurrenceTable[{a[n]==a[n-1]*(1 + a[n-1]/a[n-3]), a[0]==1, a[1]==1, a[2]==1},a,{n,0,12}] (* Vaclav Kotesovec, Jan 18 2015 *)

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

0 = a(n)*(a(n+2) - a(n+3)) + a(n+2)*a(n+2) for all n>=0.
A007660(n+1) = a(n)/a(n-1).
a(n) ~ b * f^(d^n), where b = 0.270887790039424376..., f = c^(2+sqrt(5)) = 1.574173161904651669837597516422779594... and d = (1+sqrt(5))/2. For the constant c = A258113 = 1.11305797590293193285359770716758491... see A007660. - Vaclav Kotesovec, Jan 18 2015

A121258 a(n) = a(n-1)a(n-2)a(n-3) - 1 with a(0)=a(1)=a(2)=2.

Original entry on oeis.org

2, 2, 2, 7, 27, 377, 71252, 725274107, 19482315963330427, 1006792136061113006060577048627

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..14 (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-2)* Self(n-3)-1: n in [1..12]]; // Vincenzo Librandi, Nov 14 2011

RecurrenceTable[{a[0]==a[1]==a[2]==2, a[n] == a[n-1]*a[n-2]*a[n-3] - 1}, a, {n, 0, 15}] (* G. C. Greubel, Jun 07 2019 *)
nxt[{a_,b_,c_}]:={b,c,a*b*c-1}; NestList[nxt,{2,2,2},10][[All,1]] (* Harvey P. Dale, Jun 25 2020 *)

a(n) = if(n<3, 2, a(n-1)*a(n-2)*a(n-3) - 1);
vector(12, 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-2)*a(n-3) - 1
[a(n) for n in (0..12)] # G. C. Greubel, Jun 07 2019

a(n) ~ c^(A058265^n), where c = 1.3319334322065642848267... - Vaclav Kotesovec, Jun 15 2019

Data corrected by Vincenzo Librandi, Nov 14 2011

A133400 a(0)=a(1)=a(2) = 1, thereafter a(n) = a(n-1)a(n-2)a(n-3) + 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 7, 43, 904, 272105, 10577265561, 2601826668310218121, 7488387181338771882437732599874506, 206081999881071045385328009597554265108557649484947339933019787

Offset: 0

Jonathan Vos Post, Nov 24 2007, Nov 26 2007

A tribonacci analog of A001056.

a(13) has 115 digits. - R. J. Mathar, Dec 10 2007

			a(8) = a(7)*a(6)*a(5) + 1 = 904 * 43 * 7 + 1 = 272105.
a(9) ~ 2.60182667 * 10^18.
a(10) ~ 7.48838719 * 10^33.
a(11) ~ 2.06082 * 10^62.

Seiichi Manyama, Table of n, a(n) for n = 0..16

Cf. A001056, A007660, A058265.

a:=[1,1,1];; for n in [4..15] do a[n]:=a[n-1]*a[n-2]*a[n-3]+1; od; a; # G. C. Greubel, Sep 20 2019

I:=[1,1,1]; [n le 3 select I[n] else Self(n-1)*Self(n-2)* Self(n-3) + 1: n in [1..15]]; // G. C. Greubel, Sep 20 2019

A133400 := proc(n) local i ; if n <= 2 then 1; else 1+mul( A133400(i),i=n-3..n-1) ; fi ; end: seq(A133400(n),n=0..15) ; # R. J. Mathar, Dec 10 2007

RecurrenceTable[{a[0]==1,a[1]==1,a[2]==1, a[n] == a[n-1]*a[n-2]*a[n-3] + 1},a,{n,0,15}] (* Vaclav Kotesovec, May 05 2015 *)
nxt[{a_,b_,c_}]:={b,c,a*b*c+1}; NestList[nxt,{1,1,1},15][[All,1]] (* Harvey P. Dale, Mar 05 2017 *)

m=15; v=concat([1,1,1], vector(m-3)); for(n=4, m, v[n]=v[n-1]*v[n-2] *v[n-3] +1 ); v \\ G. C. Greubel, Sep 20 2019

def a(n):
    if (n<3): return 1
    else: return a(n-1)*a(n-2)*a(n-3) + 1
[a(n) for n in (0..15)] # G. C. Greubel, Sep 20 2019

a(n) ~ c^(t^n), where c = 1.1004451797920944914628..., t = A058265 = 1.8392867552141611325518... . - Vaclav Kotesovec, May 05 2015

More terms from R. J. Mathar, Dec 10 2007

A253853 a(n) = 1 + a(n-2)*a(n-3), with a(0) = a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 7, 16, 36, 113, 577, 4069, 65202, 2347814, 265306939, 153082168429, 622891345681347, 40613761521380428832, 95353557892558423217593864, 25297960567233966143149250083396705, 3872666660463510383775257066365338059531886849

Offset: 0

Michael Somos, Jan 17 2015

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..31

Cf. A007660.

a253853 n = a253853_list !! n
a253853_list = 1 : 1 : 1 : map (+ 1)
                           (zipWith (*) a253853_list $ tail a253853_list)
-- Reinhard Zumkeller, Jan 17 2015

I:=[1,1,1]; [n le 3 select I[n] else 1 + Self(n-2)*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jan 22 2015

RecurrenceTable[{a[n]==1+a[n-2]*a[n-3],a[0]==1,a[1]==1,a[2]==1},a,{n,0,20}] (* Vaclav Kotesovec, Jan 17 2015 *)

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

a(n+5) == a(n) (mod 2) for all n>=0.
a(n+7) == a(n) (mod 7) for all n>=7.
a(n) ~ c^(d^n), where c = 1.33114442478885300080049... and d = ((27 - 3*sqrt(69)) / 2)^(1/3) / 3 + ((9 + sqrt(69))/2)^(1/3) / 3^(2/3) = 1.324717957244746... is the root of the equation d^3 = d + 1. - Vaclav Kotesovec, Jan 17 2015

A089983 1, 1, 1, 1, ... a, b, c, d, ab-cd, ...

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, 1, 2, -3, 5, 17, -91, 1532, 139497, -213710951, 29812036392235, 6371158648631364574889, -189937213493701003981668660072118562, 1210120120447335073097142485947209203511752911347585124133

Offset: 1

Ray Chandler, following a suggestion of Rainer Rosenthal, Nov 18 2003

Inspired by the formula for the determinant of a 2 X 2 matrix.

Sequence b(n,p) = a(n) (mod p), p prime, is a periodic sequence. Letting l(p) denotes the length of the period of b(n,p) we get l(2)=5, l(3)=11, l(5)=31... Is there any rule for l(p) ? - Benoit Cloitre, Nov 19 2003

Harvey P. Dale, Table of n, a(n) for n = 1..26

Cf. A089984.

I:=[1,1,1,1]; [n le 4 select I[n] else -Self(n-1)*Self(n-2)+Self(n-3)*Self(n-4): n in [1..22]]; // Vincenzo Librandi, Mar 30 2014

nxt[{a_,b_,c_,d_}]:={b,c,d,a b-c d}; NestList[nxt,{1,1,1,1},20][[All,1]] (* Harvey P. Dale, Oct 30 2021 *)

a=b=c=d=1;for(n=5,20,e=a*b-c*d;a=b;b=c;c=d;d=e;print1(e,","))

a(1)=a(2)=a(3)=a(4)=1, for n>4 a(n)=a(n-4)*a(n-3)-a(n-2)*a(n-1).

a(n) is asymptotic (in absolute value) to A^(phi^n) where phi=golden ratio and A=1.005384.. (follows same kind of behavior as A000301, A007660) - Benoit Cloitre, Nov 19 2003

A142471 a(0) = a(1) = 0; thereafter a(n) = a(n-1)*a(n-2) + 2.

Original entry on oeis.org

0, 0, 2, 2, 6, 14, 86, 1206, 103718, 125083910, 12973452977382, 1622770224612082123622, 21052933202100473722674133293917606, 34164073141115747076263787631563122725393126176374288934

Offset: 0

N. J. A. Sloane, based on email from Carla J. Garner-Bennett, Nov 13 2008

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..19
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!)

Cf. A000058, A007660, A143684.

I:=[0,0]; [n le 2 select I[n] else Self(n-1)*Self(n-2)+2: n in [1..15]]; // Vincenzo Librandi, Nov 14 2011

a:= proc(n) option remember;
      if n<2 then 0
    else a(n-1)*a(n-2) + 2
      fi; end:
seq(a(n), n=0..15); # G. C. Greubel, Apr 03 2021

a[0] = a[1] = 0; a[n_] := a[n-1] a[n-2] + 2; Table[a[n], {n, 0, 15}] (* T. D. Noe, Nov 14 2011 *)

def a(n): return 0 if n<2 else a(n-1)*a(n-2) + 2
[a(n) for n in (0..15)] # G. C. Greubel, Apr 03 2021

a(n) ~ c^(phi^n), where c = 1.278178162398588325773605473403497130099080978627235683548955136178125... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 21 2015

A143684 a(0) = a(1) = 0; thereafter a(n) = 2a(n-1)a(n-2) + 1.

Original entry on oeis.org

0, 0, 1, 1, 3, 7, 43, 603, 51859, 62541955, 6486726488691, 811385112306041061811, 10526466601050236861337066646958803, 17082036570557873538131893815781561362696563088187144467, 359626974875792367278553795120318710475396935851854517275793126801351587742904492716786003

Offset: 0

N. J. A. Sloane, based on email from Carla J. Garner-Bennett, Nov 13 2008

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..19
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!)

Cf. A000058, A007660, A142471.

I:=[0,0]; [n le 2 select I[n] else 2*Self(n-1)*Self(n-2)+1: n in [1..15]]; // Vincenzo Librandi, Nov 14 2011

a[n_]:=a[n]=If[n<2,0, 2*a[n-1]*a[n-2]+1]; Table[a[n], {n,0,15}] (* G. C. Greubel, May 29 2021 *)

def a(n): return 0 if (n<2) else 2*a(n-1)*a(n-2) + 1
[a(n) for n in (0..10)] # G. C. Greubel, May 29 2021

Equals A142471/2.

a(n) is about 1/2*c^(phi^n), where c = 1.27817816239858832577... and phi is the golden ratio. - Charles R Greathouse IV, Mar 21 2012, corrected by Vaclav Kotesovec, May 05 2015

cp's OEIS Frontend

A258113 Decimal expansion of a constant related to A007660.

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A333974 Eventual period of A007660 modulo n.

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A055937 a(n) = a(n-1) * a(n-2) - 1.

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A250309 a(n) = a(n-1)*(1 + a(n-1)/a(n-3)), with a(0) = a(1) = a(2) = 1.

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A121258 a(n) = a(n-1)*a(n-2)*a(n-3) - 1 with a(0)=a(1)=a(2)=2.

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A133400 a(0)=a(1)=a(2) = 1, thereafter a(n) = a(n-1)*a(n-2)*a(n-3) + 1.

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A253853 a(n) = 1 + a(n-2)*a(n-3), with a(0) = a(1) = a(2) = 1.

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A121258 a(n) = a(n-1)a(n-2)a(n-3) - 1 with a(0)=a(1)=a(2)=2.

A133400 a(0)=a(1)=a(2) = 1, thereafter a(n) = a(n-1)a(n-2)a(n-3) + 1.

A143684 a(0) = a(1) = 0; thereafter a(n) = 2a(n-1)a(n-2) + 1.