A005246 -id:A005246 - cp's OEIS frontend

A256045 Triangle read by rows: order of all-2s configuration on the n X k sandpile grid graph.

2, 3, 1, 7, 7, 8, 11, 5, 71, 3, 26, 9, 679, 77, 52, 41, 13, 769, 281, 17753, 29, 97, 47, 3713, 4271, 726433, 434657, 272, 153, 17, 8449, 2245, 33507, 167089, 46069729, 901, 362, 123, 81767, 8569, 24852386, 265721, 8118481057, 190818387, 73124, 571, 89, 93127, 18061, 20721019, 4213133, 4974089647, 1031151241, 1234496016491, 89893

Offset: 1

N. J. A. Sloane, Mar 15 2015

			Triangle begins:
[2]
[3, 1]
[7, 7, 8]
[11, 5, 71, 3]
[26, 9, 679, 77, 52]
[41, 13, 769, 281, 17753, 29]
[97, 47, 3713, 4271, 726433, 434657, 272]
[153, 17, 8449, 2245, 33507, 167089, 46069729, 901]
[362, 123, 81767, 8569, 24852386, 265721, 8118481057, 190818387, 73124]
[571, 89, 93127, 18061, 20721019, 4213133, 4974089647, 1031151241, 1234496016491, 89893]
...

Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter and Tianyuan Xu, Sandpiles and Dominos, Electronic Journal of Combinatorics, Volume 22(1), 2015.
David Perkinson, Lecture 15: Sandpiles, PCMI 2008 Undergraduate Summer School.

Main diagonal gives A256046, A256043, and A256047.

Cf. A005246, A195549, A348566.

From Andrey Zabolotskiy, Oct 22 2021: (Start)
It seems that T(k, 1) = A005246(k+2).
For the formula for T(k, 2), see the last theorem of Morar and Perkinson in Perkinson's slides. In particular, T(2*k, 2) = A195549(k).
T(n, k) divides A348566(n, k). (End)

Column 1 added by Andrey Zabolotskiy, Oct 22 2021

A076736 Let u(1) = u(2) = u(3) = 2, u(n) = (1 + u(n-1)*u(n-2))/u(n-3); then a(n) is the denominator of u(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 2, 8, 4, 16, 8, 32, 16, 64, 32, 128, 64, 256, 128, 512, 256, 1024, 512, 2048, 1024, 4096, 2048, 8192, 4096, 16384, 8192, 32768, 16384, 65536, 32768, 131072, 65536, 262144, 131072, 524288, 262144, 1048576, 524288, 2097152

Offset: 1

Benoit Cloitre, Nov 24 2002

The sequence 1,4,2,8,4,... has g.f. (1+4*x)/(1-2*x^2) and a(n) = 2^(n/2)*(1+2*sqrt(2) + (1-2*sqrt(2))*(-1)^n)/2. - Paul Barry, Apr 26 2004

The sequence 2,1,4,2,8,... has a(n) = 2^(n/2)*(1+2*sqrt(2)-(1-2*sqrt(2))*(-1)^n)/(2*sqrt(2)) and is essentially the pair-reversal of A016116. - Paul Barry, Apr 26 2004

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A005246, A076737.

LinearRecurrence[{0,2},{1,1,1,2,1},50] (* Harvey P. Dale, Aug 25 2015 *)

For n > 4, a(n) = 2^A028242(n-4).
From Colin Barker, Oct 14 2014: (Start)
For n > 5, a(n) = 2*a(n-2).
G.f.: x*(x-1)*(x^3+x^2+2*x+1) / (2*x^2-1). (End)

More terms from Paul Barry, Apr 26 2004

A208207 a(n)=(a(n-1)^3*a(n-2)+1)/a(n-3) with a(0)=a(1)=a(2)=1.

Original entry on oeis.org

1, 1, 1, 2, 9, 1459, 13975855106, 442535332406378982945622818194705

Offset: 0

Matthew C. Russell, Apr 23 2012

nonn

This is the case a=1, b=3, 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 has 105 digits. - Harvey P. Dale, Jul 04 2022

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, A208206, A208208, A208213.

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

RecurrenceTable[{a[n] == (a[n - 1]^3*a[n - 2] + 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^3 b + 1)/a}; NestList[nxt,{1,1,1},10][[All,1]] (* Harvey P. Dale, Jul 04 2022 *)

From Vaclav Kotesovec, May 20 2015: (Start)
a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where
d1 = -0.675130870566646070889621798150060480808032527677372732612153869841...
d2 = 0.4608111271891108834741240973014799919001128904578732982807715533323...
d3 = 3.2143197433775351874154977008485804889079196372194994343313823165091...
are the roots of the equation d^3 + 1 = 3*d^2 + d and
c1 = 0.8399660110229591295951614867364338523629139731316529610703364786466...
c2 = 0.5166029105674572719002224224720428001985297645051505025129589573676...
c3 = 1.0214282112585594227681235564690028577352359049566082298453239674712...
(End)

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

Original entry on oeis.org

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)

A211956 Coefficients of a sequence of polynomials related to the Morgan-Voyce polynomials.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 4, 2, 1, 6, 4, 1, 9, 12, 4, 1, 12, 20, 8, 1, 16, 40, 32, 8, 1, 20, 60, 56, 16, 1, 25, 100, 140, 80, 16, 1, 30, 140, 224, 144, 32, 1, 36, 210, 448, 432, 192, 32, 1, 42, 280, 672, 720, 352, 64, 1, 49, 392, 1176, 1680, 1232, 448, 64

Offset: 0

Peter Bala, Apr 30 2012

The row generating polynomials R(n,x) of A211955 factorize in the ring Z[x] as R(n,x) = P(n,x)*P(n+1,x) for n >= 1: explicitly, P(2*n,x) = 1/2*(b(2*n,2*x) + 1)/b(n,2*x) and P(2*n+1,x) = b(n,2*x), where b(n,x) := Sum_{k = 0..n} binomial(n+k,2*k)*x^k are the Morgan-Voyce polynomials of A085478. This triangle lists the coefficients in ascending powers of x of the polynomials P(n,x).

The odd numbered rows of the present triangle produce triangle A123519; the even numbered row entries are recorded separately in A211957 and appear to equal the unsigned and row reversed form of A204021. The even numbered rows with a factor of 2^(k-1) removed from the k-th column entries produce triangle A208513.

			Triangle begins
.n\k.|..0....1....2....3....4
= = = = = = = = = = = = = = =
..0..|..1
..1..|..1
..2..|..1....1
..3..|..1....2
..4..|..1....4....2
..5..|..1....6....4
..6..|..1....9...12....4
..7..|..1...12...20....8
..8..|..1...16...40...32....8
..9..|..1...20...60...56...16
...

Eric Weisstein's World of Mathematics, Morgan-Voyce polynomials

Cf. A005246 (row sums), A085478, A123519, A204021, A208513, A211955, A211957.

T(n,0) = 1; for k > 0, T(2*n,k) = 2^k * binomial(n+k,2*k) = A123519(n,k);
for k > 0, T(2*n-1,k) = n/(n+k)*(2^k)*binomial(n+k,2*k) = 2^(k-1)*A208513(n,k).
O.g.f.: ((1+t)*(1-t^2)-t^2*x)/((1-t^2)^2-2*t^2*x) = 1 + t + (1+x)*t^2 + (1+2*x)*t^3 + (1+4*x+2*x^2)*t^4 + ....
Row generating polynomials: P(2*n,x) := 1/2*(b(2*n,2*x)+1)/b(n,2*x) and P(2*n+1,x) := b(n,2*x), where b(n,x) := Sum_{k = 0..n} binomial(n+k,2*k)*x^k are the Morgan-Voyce polynomials of A085478.
The product P(n,x)*P(n+1,x) is the n-th row polynomial of A211955.
In terms of T(n,x), the Chebyshev polynomials of the first kind, we have P(2*n,x) = T(2*n,u) and P(2*n+1,x) = 1/u*T(2*n+1,u), where u = sqrt((x+2)/2).
Other representations for the row polynomials include
P(2*n,x) = 1/2*(1+x+sqrt(x^2+2*x))^n + 1/2*(1+x-sqrt(x^2+2*x))^n;
P(2*n,x) = n*Sum_{k = 0..n}(-1)^(n-k)/(n+k) * binomial(n+k,2*k) * (2*x+4)^k for n >= 1;
P(2*n+1,x) = (2*n+1)*Sum_{k=0..n} (-1)^(n-k)/(n+k+1) * binomial(n+k+1,2*k+1) * (2*x+4)^k.
Recurrence equation: P(n+1,x)*P(n-2,x) - P(n,x)*P(n-1,x) = x.
Row sums A005246(n+2).

A143643 Numerators of the lower principal convergents and the lower intermediate convergents to 3^(1/2).

Original entry on oeis.org

1, 3, 5, 12, 19, 45, 71, 168, 265, 627, 989, 2340, 3691, 8733, 13775, 32592, 51409, 121635, 191861, 453948, 716035, 1694157, 2672279, 6322680, 9973081, 23596563, 37220045, 88063572, 138907099, 328657725, 518408351, 1226567328, 1934726305, 4577611587, 7220496869, 17083879020, 26947261171, 63757904493, 100568547815

Offset: 1

Clark Kimberling, Aug 27 2008

The lower principal and intermediate convergents to 3^(1/2), beginning with 1/1, 3/2, 5/3, 12/7, 19/11, form a strictly increasing sequence; with essentially, numerators being this sequence and denominators being A005246.
sqrt(floor(a(n)^2/3)+1) = A005246(n+1). Also see A082630. - Richard R. Forberg, Nov 14 2013
a(n) = U_n(sqrt(6),1) for n odd and a(n) = 3*U_n(sqrt(6),1) for n even, where U_n(sqrt(R),Q) denotes the Lehmer sequence with parameters R and Q. This sequence is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this sequence is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Sep 03 2019

			From _Peter Bala_, Sep 03 2019: (Start)
If p(n)/q(n) denotes the n-th convergent to the simple continued fraction alpha = [c(0); c(1), c(2), ...] then a lower semiconvergent is a rational number of the form ( p(2*n) + m*p(2*n+1) )/( q(2*n) + m*q(2*n+1) ) where 0 <= m <= c(2*n+2). The lower semiconvergents include the even-indexed convergents p(2*n)/q(2*n) and give an increasing sequence of approximations to alpha from below.
In this case the simple continued fraction expansion sqrt(3) = [1; 1, 2, 1, 2, ...] produces the sequence of convergents (p(n)/q(n))n>=0 = [1/1, 2/1, 5/3, 7/4, 19/11, 26,15, 71/41, ...].
Thus the increasing sequence of lower semiconvergents begins 1/1, (1 + 2)/(1 + 1) = 3/2, (1 + 2*2)/(1 + 2*1) = 5/3, (5 + 7)/(3 + 4) = 12/7, (5 + 2*7)/(3 + 2*4) = 19/11, ... with numerators 1, 3, 5, 12, 19, .... (End)

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.
Clark Kimberling, "Best lower and upper approximates to irrational numbers," Elemente der Mathematik, 52 (1997) 122-126.

Wikipedia, Lehmer sequence

Cf. A002530, A002531, A005246, A082630, A143607, A143609, A143642.

a(n) = 4*a(n-2)-a(n-4). G.f.: x*(1+3*x+x^2)/(1-4*x^2+x^4). - Colin Barker, Apr 28 2012

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)

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

Original entry on oeis.org

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)

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)

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)

cp's OEIS Frontend

A256045 Triangle read by rows: order of all-2s configuration on the n X k sandpile grid graph.

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A076736 Let u(1) = u(2) = u(3) = 2, u(n) = (1 + u(n-1)*u(n-2))/u(n-3); then a(n) is the denominator of u(n).

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A208207 a(n)=(a(n-1)^3*a(n-2)+1)/a(n-3) with a(0)=a(1)=a(2)=1.

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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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A211956 Coefficients of a sequence of polynomials related to the Morgan-Voyce polynomials.

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A143643 Numerators of the lower principal convergents and the lower intermediate convergents to 3^(1/2).

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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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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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