A208203 -id:A208203 - cp's OEIS frontend

A208210 a(n)=(a(n-1)^2*a(n-2)^3+1)/a(n-3) with a(0)=a(1)=a(2)=1.

1, 1, 1, 2, 5, 201, 2525063, 10355298070412763074, 8589063344901709900442551790362661608528200120823830773

Offset: 0

Matthew C. Russell, Apr 23 2012

nonn

This is the case a=3, 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 161 digits. - Harvey P. Dale, Apr 14 2022

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).
Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, Advances in Applied Mathematics 28 (2002), 119-144.

Cf. A005246, A208203, A208209, A208211, A208214.

a:=proc(n) if n<3 then return 1: fi: return (a(n-1)^2*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]^2*a[n-2]^3 + 1)/a[n-3];
Array[a, 10, 0] (* Jean-François Alcover, Dec 14 2017 *)
nxt[{a_,b_,c_}]:={b,c,(c^2 b^3+1)/a}; NestList[nxt,{1,1,1},10][[All,1]] (* Harvey P. Dale, Apr 14 2022 *)

From Vaclav Kotesovec, May 20 2015: (Start)
a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where
d1 = -1.198691243515997113071999692569776193916276872472594369204332359716...
d2 = 0.2864620650316004980582127604312427653427138786836169481458128553091...
d3 = 2.9122291784843966150137869321385334285735629937889774210585195044073...
are the roots of the equation d^3 + 1 = 2*d^2 + 3*d and
c1 = 0.9326266928252752296152676800592959458631493222642463226349218269187...
c2 = 0.2535475214701961189033928082745089316567819534655391761010907360554...
c3 = 1.0248087086665041891835364490857429725941144848712661648932932629036...
(End)

A208202 a(n) = (a(n-1)*a(n-2)^2+1)/a(n-3) with a(0)=a(1)=a(2)=1.

Original entry on oeis.org

1, 1, 1, 2, 3, 13, 59, 3324, 890065, 166683166499, 39725939269090918399, 1240040687243304530118746458657221560, 11740660815927242416329935330676365456512664243108711550072429939

Offset: 0

Matthew C. Russell, Apr 23 2012

nonn

This is the case a=2, 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).

Bruno Berselli, Table of n, a(n) for n = 0..16
Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, arXiv:math/0104241v1 [math.CO] (2001), Advances in Applied Mathematics 28 (2002), 119-144.

Cf. A005246, A208203, A208209.

[n le 3 select 1 else (Self(n-1)*Self(n-2)^2+1)/Self(n-3): n in [1..13]]; // Bruno Berselli, Apr 24 2012

RecurrenceTable[{a[0] == a[1] == a[2] == 1, a[n] == (a[n - 1] a[n - 2]^2 + 1)/a[n - 3]}, a, {n, 12}] (* Bruno Berselli, Apr 25 2012 *)
(* The numerical values of the constants d1, d2, d3 *) Print[N[{Root[1-2*#1-#1^2+#1^3&,1], Root[1-2*#1-#1^2+#1^3&,2], Root[1-2*#1-#1^2+#1^3&,3]}, 80]]; (* and the constants c1, c2, c3 *) A208202 = RecurrenceTable[{a[0]==a[1]==a[2]==N[1,100], a[n] == (a[n-1]*a[n-2]^2 + 1)/a[n-3]}, a, {n,1,30}]; Table[Flatten[N[{Exp[cc1], Exp[cc2], Exp[cc3]}/.Solve[Table[Log[A208202[[n]]] == cc1*Root[1 - 2*#1 - #1^2 + #1^3&, 1]^n + cc2*Root[1 - 2*#1 - #1^2 + #1^3&, 2]^n + cc3*Root[1 - 2*#1 - #1^2 + #1^3&, 3]^n, {n, k, k+2}]],80]], {k, Length[A208202]-3, Length[A208202]-2}] (* Vaclav Kotesovec, May 20 2015 *)

From Vaclav Kotesovec, May 20 2015: (Start)
a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where
d1 = -1.24697960371746706105000976800847962126454946179280421073109887819...
d2 = 0.445041867912628808577805128993589518932711137529089910623974031794...
d3 = 1.801937735804838252472204639014890102331838324263714300107124846398...
are the roots of the equation d^3 + 1 = d^2 + 2*d and
c1 = 0.937508205283971584227188160392119895660526011507051773879367647962...
c2 = 0.127128212809518009874462927372545164747593272064601714573478901156...
c3 = 1.135040592200579625529345655593495454581148721169010026906480955795...
(End)

A208204 a(n) = (a(n-1)*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, 3, 49, 1985, 3814376662, 1208563686390770296199, 128885284912846137074628029815898112630258374651779168689

Offset: 0

Matthew C. Russell, Apr 23 2012

nonn

This is the case a=4, 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..12
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, A208203, A208205, A208211.

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

a[0] = a[1] = a[2] = 1; a[n_] := a[n] = (a[n - 1]*a[n - 2]^4 + 1)/a[n - 3];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Dec 14 2017 *)

From Vaclav Kotesovec, May 20 2015: (Start)
a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where
d1 = -1.699628148275317956229728291667145232598924547592878096541472700997...
d2 = 0.2391232782565544642500835033134825869161430421361867747730632704531...
d3 = 2.4605048700187634919796447883536626456827815054566913217684094305444...
are the roots of the equation d^3 + 1 = d^2 + 4*d and
c1 = 0.9668824482256124500532459849115781952211866063916062435395239896336...
c2 = 0.0680423294122660088493946488133224274885942757072304155092839505634...
c3 = 1.0386083844527725102069795872299989830277012965629707721463998933768...
(End)

cp's OEIS Frontend

A208210 a(n)=(a(n-1)^2*a(n-2)^3+1)/a(n-3) with a(0)=a(1)=a(2)=1.

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A208202 a(n) = (a(n-1)*a(n-2)^2+1)/a(n-3) with a(0)=a(1)=a(2)=1.

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A208204 a(n) = (a(n-1)*a(n-2)^4+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