A157014 -id:A157014 - cp's OEIS frontend

A157460 Expansion of 88x^2 / (1-483x+483*x^2-x^3).

0, 88, 42504, 20486928, 9874656880, 4759564129320, 2294100035675448, 1105751457631436704, 532969908478316815968, 256890390135091073859960, 123820635075205419283684840, 59681289215858877003662233008, 28766257581408903510345912625104

Offset: 1

Paul Weisenhorn, Mar 01 2009

This sequence is part of a solution of a more general problem involving two equations, three sequences a(n), b(n), c(n) and a constant A:
    A    * c(n)+1 = a(n)^2,
   (A+1) * c(n)+1 = b(n)^2, for details see comment in A157014.
A157460 is the c(n) sequence for A=5.

Colin Barker, Table of n, a(n) for n = 1..350
Index entries for linear recurrences with constant coefficients, signature (483,-483,1).

5*A157460(n)+1 = A157014(n)^2 for n>=1.

6*A157460(n)+1 = A133283(n)^2 for n>=1.

CoefficientList[Series[88x^2/(1-483x+483x^2-x^3),{x,0,30}],x] (* or *) LinearRecurrence[{483,-483,1},{0,0,88},30] (* Harvey P. Dale, Apr 16 2015 *)

concat(0, Vec(88*x^2/(1-483*x+483*x^2-x^3)+O(x^20))) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = round(-((241+44*sqrt(30))^(-n)*(-1+(241+44*sqrt(30))^n)*(11+2*sqrt(30)+(-11+2*sqrt(30))*(241+44*sqrt(30))^n))/120) \\ Colin Barker, Jul 25 2016

G.f.: 88*x^2 / (1-483*x+483*x^2-x^3).
c(1) = 0, c(2) = 88, c(3) = 483*c(2), c(n) = 483*(c(n-1)-c(n-2))+c(n-3) for n>3.
a(n) = -((241+44*sqrt(30))^(-n)*(-1+(241+44*sqrt(30))^n)*(11+2*sqrt(30)+(-11+2*sqrt(30))*(241+44*sqrt(30))^n))/120. - Colin Barker, Jul 25 2016

Edited by Alois P. Heinz, Sep 09 2011

A157880 Expansion of 136x^2 / (-x^3+1155x^2-1155*x+1).

Original entry on oeis.org

0, 136, 157080, 181270320, 209185792336, 241400223085560, 278575648254944040, 321476056685982336736, 370983090839975361649440, 428114165353274881361117160, 494043375834588373115367553336, 570125627598949629300252795432720, 657924480205812037624118610561805680

Offset: 1

Paul Weisenhorn, Mar 08 2009

This sequence is part of a solution of a more general problem involving two equations, three sequences a(n), b(n), c(n) and a constant A:
    A    * c(n)+1 = a(n)^2,
   (A+1) * c(n)+1 = b(n)^2, for details see comment in A157014.
A157880 is the c(n) sequence for A=8.

Colin Barker, Table of n, a(n) for n = 1..325
Index entries for linear recurrences with constant coefficients, signature (1155,-1155,1).

8*A157880(n)+1 = A077420(n-1)^2.

9*A157880(n)+1 = A046176(n)^2.

LinearRecurrence[{1155,-1155,1},{0,136,157080},20] (* Harvey P. Dale, Dec 04 2019 *)

concat(0, Vec(136*x^2/(-x^3+1155*x^2-1155*x+1) + O(x^20))) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = round(-((577+408*sqrt(2))^(-n)*(-1+(577+408*sqrt(2))^n)*(17+12*sqrt(2)+(-17+12*sqrt(2))*(577+408*sqrt(2))^n))/288) \\ Colin Barker, Jul 25 2016

G.f.: 136*x^2/(-x^3+1155*x^2-1155*x+1).
c(1) = 0, c(2) = 136, c(3) = 1155*c(2), c(n) = 1155 * (c(n-1)-c(n-2)) + c(n-3) for n>3.
a(n) = -((577+408*sqrt(2))^(-n)*(-1+(577+408*sqrt(2))^n)*(17+12*sqrt(2)+(-17+12*sqrt(2))*(577+408*sqrt(2))^n))/288. - Colin Barker, Jul 25 2016

Edited by Alois P. Heinz, Sep 09 2011

A157881 Expansion of 152x^2 / (-x^3+1443x^2-1443*x+1).

Original entry on oeis.org

0, 152, 219336, 316282512, 456079163120, 657665836936680, 948353680783529592, 1367525350024012735136, 1971970606380945580536672, 2843580246875973503121146040, 4100440744024547410555112053160, 5912832709303150490046968459510832

Offset: 1

Paul Weisenhorn, Mar 08 2009, Jun 25 2009

This sequence is part of a solution of a more general problem involving two equations, three sequences a(n), b(n), c(n) and a constant A:
    A    * c(n)+1 = a(n)^2,
   (A+1) * c(n)+1 = b(n)^2, for details see comment in A157014.
A157881 is the c(n) sequence for A=9.

Colin Barker, Table of n, a(n) for n = 1..300
Index entries for linear recurrences with constant coefficients, signature (1443,-1443,1).

8*A157881(n)+1 = A097315(n-1)^2.

9*A157881(n)+1 = A097314(n-1)^2.

LinearRecurrence[{1443,-1443,1},{0,152,219336},20] (* Harvey P. Dale, Jul 18 2019 *)

concat(0, Vec(152*x^2/(-x^3+1443*x^2-1443*x+1) + O(x^20))) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = round(-((721+228*sqrt(10))^(-n)*(-1+(721+228*sqrt(10))^n)*(19+6*sqrt(10)+(-19+6*sqrt(10))*(721+228*sqrt(10))^n))/360) \\ Colin Barker, Jul 25 2016

G.f.: 152*x^2/(-x^3+1443*x^2-1443*x+1).
c(1) = 0, c(2) = 152, c(3) = 1443*c(2), c(n) = 1443 * (c(n-1)-c(n-2)) + c(n-3) for n>3.
a(n) = -((721+228*sqrt(10))^(-n)*(-1+(721+228*sqrt(10))^n)*(19+6*sqrt(10)+(-19+6*sqrt(10))*(721+228*sqrt(10))^n))/360. - Colin Barker, Jul 25 2016

Edited by Alois P. Heinz, Sep 09 2011

A238245 Positive integers n such that x^2 - 22xy + y^2 + n = 0 has integer solutions.

Original entry on oeis.org

20, 39, 56, 71, 80, 84, 95, 104, 111, 116, 119, 120, 156, 180, 191, 224, 239, 255, 284, 296, 311, 320, 336, 351, 359, 380, 399, 404, 416, 431, 444, 455, 464, 471, 476, 479, 480, 500, 504, 551, 596, 599, 624, 639, 680, 695, 696, 719, 720, 756, 764, 791, 824

Offset: 1

Colin Barker, Feb 20 2014

nonn

			39 is in the sequence because x^2 - 22xy + y^2 + 39 = 0 has integer solutions, for example (x, y) = (2, 43).

Cf. A157014 (n = 20), A137881 (n = 104), A077422 (n = 120), A133275 (n = 180).

Cf. A031363, A084917, A237351, A237599, A237606, A237609, A237610, A236330, A236331, A238240.

A269028 a(n) = 40*a(n - 1) - a(n - 2) for n>1, a(0) = 1, a(1) = 1.

Original entry on oeis.org

1, 1, 39, 1559, 62321, 2491281, 99588919, 3981065479, 159143030241, 6361740144161, 254310462736199, 10166056769303799, 406387960309415761, 16245352355607326641, 649407706263983649879, 25960062898203738668519, 1037753108221885563090881

Offset: 0

Ilya Gutkovskiy, Feb 18 2016

In general, the ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - b(n - 2) with n>1 and b(0)=1, b(1)=1, is (1 - (k - 1)*x)/(1 - k*x +x^2). This recurrence gives the closed form b(n) = (2^( -n - 1)*((k - 2)*(k - sqrt(k^2 - 4))^n + sqrt(k^2 - 4)*(k - sqrt(k^2 - 4))^n - (k - 2)*(sqrt(k^2 - 4) + k)^n + sqrt(k^2 - 4)*(sqrt(k^2 - 4) + k)^n))/sqrt(k^2 - 4).

Index entries for linear recurrences with constant coefficients, signature (40,-1).

Cf. A001519, A001835, A001653, A049685, A070997, A070998, A072256, A078922, A160682, A007805, A075839, A157014, A159664, A159668, A157877, A238379, A097315.

[n le 2 select 1 else 40*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 19 2016

Table[Cosh[n Log[20 + Sqrt[399]]] - Sqrt[19/21] Sinh[n Log[20 + Sqrt[399]]], {n, 0, 17}]
Table[(2^(-n - 2) (38 (40 - 2 Sqrt[399])^n + 2 Sqrt[399] (40 - 2 Sqrt[399])^n - 38 (40 + 2 Sqrt[399])^n + 2 Sqrt[399] (40 + 2 Sqrt[399])^n))/Sqrt[399], {n, 0, 17}]
LinearRecurrence[{40, -1}, {1, 1}, 17]

G.f.: (1 - 39*x)/(1 - 40*x + x^2).
a(n) = cosh(n*log(20 + sqrt(399))) - sqrt(19/21)*sinh(n*log(20 + sqrt(399))).
a(n) = (2^(-n - 2)*(38*(40 - 2*sqrt(399))^n + 2*sqrt(399)*(40 - 2*sqrt(399))^n - 38*(40 + 2*sqrt(399))^n + 2*sqrt(399)*(40 + 2*sqrt(399))^n))/sqrt(399).
Sum_{n>=0} 1/a(n) = 2.0262989201139499769986...

cp's OEIS Frontend

A157460 Expansion of 88*x^2 / (1-483*x+483*x^2-x^3).

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A157880 Expansion of 136*x^2 / (-x^3+1155*x^2-1155*x+1).

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A157881 Expansion of 152*x^2 / (-x^3+1443*x^2-1443*x+1).

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A238245 Positive integers n such that x^2 - 22xy + y^2 + n = 0 has integer solutions.

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A269028 a(n) = 40*a(n - 1) - a(n - 2) for n>1, a(0) = 1, a(1) = 1.

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A157460 Expansion of 88x^2 / (1-483x+483*x^2-x^3).

A157880 Expansion of 136x^2 / (-x^3+1155x^2-1155*x+1).

A157881 Expansion of 152x^2 / (-x^3+1443x^2-1443*x+1).