A208204 -id:A208204 - cp's OEIS frontend

A208203 a(n) = (a(n-1)*a(n-2)^3+1)/a(n-3) with a(0)=a(1)=a(2)=1.

1, 1, 1, 2, 3, 25, 338, 1760417, 2719102918193, 43888992061611808973481301345, 501206842313618355048837897498360450999462416742984495192498

Offset: 0

Matthew C. Russell, Apr 23 2012

nonn

This is the case a=3, b=1, y(0)=y(1)=y(2)=1 of the recurrence shown in the Example 3.2 of "The Laurent phenomenon" (see Link lines, p. 10).

Seiichi Manyama, Table of n, a(n) for n = 0..13
Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, arXiv:math/0104241v1 [math.CO] (2001).
Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, Advances in Applied Mathematics 28 (2002), 119-144.

Cf. A005246, A208202, A208204, A208210.

a:=proc(n) if n<3 then return 1: fi: return (a(n-1)*a(n-2)^3+1)/a(n-3): end: seq(a(i),i=0..10);

a[0] = a[1] = a[2] = 1; a[n_] := a[n] = (a[n - 1]*a[n - 2]^3 + 1)/a[n - 3];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Dec 14 2017 *)
nxt[{a_,b_,c_}]:={b,c,(c*b^3+1)/a}; NestList[nxt,{1,1,1},10][[;;,1]] (* Harvey P. Dale, Nov 19 2023 *)

From Vaclav Kotesovec, May 20 2015: (Start)
a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where
d1 = -1.481194304092015622633537241216857180552745216998476728395893140813...
d2 = 0.3111078174659818999302814767914862551326055871751667747271657344269...
d3 = 2.1700864866260337227032557644253709254201396298233099536687274063868...
are the roots of the equation d^3 + 1 = d^2 + 3*d and
c1 = 0.9558632550121524723294926402589664329208850973886195977958538648966...
c2 = 0.0925177857987965285678801091508493414479538300221910521000975614673...
c3 = 1.0621981744880569938247885786471114069804924018378928906529142898259...
(End)

A208211 a(n)=(a(n-1)^2*a(n-2)^4+1)/a(n-3) with a(0)=a(1)=a(2)=1.

Original entry on oeis.org

1, 1, 1, 2, 5, 401, 50250313, 13058251494934169005517674, 2711319949800838662068317571116321157238013748056632969662193456875554487084437

Offset: 0

Matthew C. Russell, Apr 23 2012

nonn

This is the case a=4, b=2, y(0)=y(1)=y(2)=1 of the recurrence shown in the Example 3.2 of "The Laurent phenomenon" (see Link lines, p. 10).

The next term, a(9), has 250 digits. - Harvey P. Dale, May 12 2015

Seiichi Manyama, Table of n, a(n) for n = 0..10
Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, arXiv:math/0104241v1 [math.CO] (2001), Advances in Applied Mathematics 28 (2002), 119-144.

Cf. A005246, A208204, A208210, A208212.

a:=proc(n) if n<3 then return 1: fi: return (a(n-1)^2*a(n-2)^4+1)/a(n-3): end: seq(a(i),i=0..10);

RecurrenceTable[{a[0]==a[1]==a[2]==1,a[n]==(a[n-1]^2*a[n-2]^4+1)/a[n-3]},a,{n,9}] (* Harvey P. Dale, May 12 2015 *)

From Vaclav Kotesovec, May 20 2015: (Start)
a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where
d1 = -1.391382380630900845100729034616031832171938259539254240563846155543...
d2 = 0.2271344421706896320468868758105588761186297860618178147525916240716...
d3 = 3.1642479384602112130538421588054729560533084734774364258112545314714...
are the roots of the equation d^3 + 1 = 2*d^2 + 4*d and
c1 = 0.9492747639156309053009206968548726546571223067568220073025225799006...
c2 = 0.2025736158012536053359109009272747757676200151893348144191432397054...
c3 = 1.0182066570849459786725527422494583474915007718333213073686225606760...
(End)

A208205 a(n)=(a(n-1)*a(n-2)^5+1)/a(n-3) with a(0)=a(1)=a(2)=1.

Original entry on oeis.org

1, 1, 1, 2, 3, 97, 11786, 33736797423001, 79097781524295318019203322936641

Offset: 0

Matthew C. Russell, Apr 23 2012

nonn

This is the case a=5, b=1, y(0)=y(1)=y(2)=1 of the recurrence shown in the Example 3.2 of "The Laurent phenomenon" (see Link lines, p. 10)

The next term (a(9)) has 96 digits. - Harvey P. Dale, Sep 13 2022

Seiichi Manyama, Table of n, a(n) for n = 0..11
Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, arXiv:math/0104241v1 [math.CO] (2001), Advances in Applied Mathematics 28 (2002), 119-144.

Cf. A005246, A208204.

a:=proc(n) if n<3 then return 1: fi: return (a(n-1)*a(n-2)^5+1)/a(n-3): end: seq(a(i),i=1..10);

RecurrenceTable[{a[n] == (a[n - 1] a[n - 2]^5 + 1)/a[n - 3], a[0] == a[1] == a[2] == 1}, a, {n, 0, 8}] (* Michael De Vlieger, Mar 19 2017 *)
nxt[{a_,b_,c_}]:={b,c,(c*b^5+1)/a}; NestList[nxt,{1,1,1},10][[All,1]] (* Harvey P. Dale, Sep 13 2022 *)

From Vaclav Kotesovec, May 20 2015: (Start)
a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where
d1 = -1.903211925911553287485216224057094233775314050044332659604216582082...
d2 = 0.1939365664746304482560845569332033002552873106788960042162607290276...
d3 = 2.7092753594369228392291316671238909335200267393654366553879558530545...
are the roots of the equation d^3 + 1 = d^2 + 5*d and
c1 = 0.9741074409555962981370572554321352591111177638556227590517984272608...
c2 = 0.0499759123576461468686480770163694779918691526759585723897652462761...
c3 = 1.0272217210627198315132544386598971884129462517962425299212701250318...
(End)

cp's OEIS Frontend

A208203 a(n) = (a(n-1)*a(n-2)^3+1)/a(n-3) with a(0)=a(1)=a(2)=1.

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A208211 a(n)=(a(n-1)^2*a(n-2)^4+1)/a(n-3) with a(0)=a(1)=a(2)=1.

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A208205 a(n)=(a(n-1)*a(n-2)^5+1)/a(n-3) with a(0)=a(1)=a(2)=1.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula