A208211 -id:A208211 - 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)

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)

A208212 a(n) = (a(n-1)^2*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, 5, 801, 1002501563, 66276977238296815913344374183794

Offset: 0

Matthew C. Russell, Apr 23 2012

nonn

This is the case a=5, 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).

Seiichi Manyama, Table of n, a(n) for n = 0..9
Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, arXiv:math/0104241v1 [math.CO] (2001); Advances in Applied Mathematics 28 (2002), 119-144.

Cf. A005246, A208211.

a:=proc(n) if n<3 then return 1: fi: return (a(n-1)^2*a(n-2)^5+1)/a(n-3): end: seq(a(i),i=0..10);

a[n_] := a[n] = If[n <= 2, 1, (a[n - 1]^2*a[n - 2]^5 + 1)/a[n - 3]];
Table[a[n], {n, 0, 7}] (* Jean-François Alcover, Nov 25 2017 *)

From Vaclav Kotesovec, May 20 2015: (Start)
a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where
d1 = -1.575773472651936072015953246349296378313356749177416595434978648425...
d2 = 0.1872837251102239188569922313039458439968721185362219238420888422761...
d3 = 3.3884897475417121531589610150453505343164846306411946715928898061494...
are the roots of the equation d^3 + 1 = 2*d^2 + 5*d and
c1 = 0.9607631794694254165284953988161129828633931861764073755339129251426...
c2 = 0.1625672201779811599302887070429221376610589038410298300467412998556...
c3 = 1.0141969317515019907302101637404918873873074910913934972790303073225...
(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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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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A208212 a(n) = (a(n-1)^2*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