A097775 -id:A097775 - cp's OEIS frontend

A097741 Pell equation solutions (10a(n))^2 - 101b(n)^2 = -1 with b(n) = A097742(n), n >= 0.

1, 403, 162005, 65125607, 26180332009, 10524428342011, 4230794013156413, 1700768668860536015, 683704774087922321617, 274847618414675912754019, 110488058897925629004794021, 44415924829347688184014442423, 17855091293338872724344801060025

Offset: 0

Wolfdieter Lang, Aug 31 2004

			(x,y) = (10*1=10;1), (4030=10*403;401), (1620050=10*162005;161201), ... give the positive integer solutions to x^2 - 101*y^2 =-1.

Indranil Ghosh, Table of n, a(n) for n = 0..383
Tanya Khovanova, Recursive Sequences
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (402,-1).

Cf. A097740 for S(n, 2*201).

Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.

LinearRecurrence[{402, -1}, {1, 403}, 20] (* or *) CoefficientList[Series[(1 + x)/(1 - 402 x + x^2), {x, 0, 20}], x]  (* Harvey P. Dale, Apr 20 2011 *)
a[n_] := Floor[(10 + Sqrt[101])^(2 n + 1)]/20;
Table[a[n], {n, 0, 11}] (* Peter Luschny, Apr 05 2018 *)

x='x+O('x^99); Vec((1+x)/(1-2*201*x+x^2)) \\ Altug Alkan, Apr 05 2018

G.f.: (1 + x)/(1 - 2*201*x + x^2).
a(n) = S(n, 2*201) + S(n-1, 2*201) = S(2*n, 2*sqrt(101)), with Chebyshev polynomials of the 2nd kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x).
a(n) = ((-1)^n)*T(2*n+1, 10*i)/(10*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.
a(n) = 402*a(n-1) - a(n-2) for n > 1, a(0)=1, a(1)=403. - Philippe Deléham, Nov 18 2008
a(n) = (1/10)*sinh((2*n + 1)*arcsinh(10)). - Bruno Berselli, Apr 03 2018
Let h = (10 + sqrt(101))^(2*n+1) then a(n) = (h-1/h)/20 and a(n) = floor(h/20). - Peter Luschny, Apr 05 2018

A097766 Pell equation solutions (11a(n))^2 - 122b(n)^2 = -1 with b(n):=A097767(n), n >= 0.

Original entry on oeis.org

1, 487, 236681, 115026479, 55902632113, 27168564180439, 13203866289061241, 6417051847919582687, 3118673994222628124641, 1515669144140349348992839, 736612085378215560982395113, 357991957824668622288095032079, 173983354890703572216453203195281

Offset: 0

Wolfdieter Lang, Aug 31 2004

			(x,y) = (11*1=11;1), (5357=11*487;485), (2603491=11*236681;235709), ... give the positive integer solutions to x^2 - 122*y^2 =-1.

Michael De Vlieger, Table of n, a(n) for n = 0..372
Tanya Khovanova, Recursive Sequences
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (486,-1).

Cf. A097765 for S(n, 2*243).

Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.

LinearRecurrence[{486, -1},{1, 487},12] (* Ray Chandler, Aug 12 2015 *)

x='x+O('x^99); Vec((1+x)/(1-2*243*x+x^2)) \\ Altug Alkan, Apr 05 2018

G.f.: (1 + x)/(1 - 2*243*x + x^2).
a(n) = S(n, 2*243) + S(n-1, 2*243) = S(2*n, 2*sqrt(122)), with Chebyshev polynomials of the second kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x).
a(n) = ((-1)^n)*T(2*n+1, 11*i)/(11*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.
a(n) = 486*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=487. - Philippe Deléham, Nov 18 2008
a(n) = (1/11)*sinh((2*n + 1)*arcsinh(11)). - Bruno Berselli, Apr 03 2018

A097769 Pell equation solutions (12a(n))^2 - 145b(n)^2 = -1 with b(n):=A097770(n), n >= 0.

Original entry on oeis.org

1, 579, 334661, 193433479, 111804216201, 64622643530699, 37351776156527821, 21589261995829549839, 12478556081813323279121, 7212583826026105025782099, 4168860972887006891578774101, 2409594429744863957227505648279

Offset: 0

Wolfdieter Lang, Aug 31 2004

			(x,y) = (12*1=12;1), (6948=12*579;577), (4015932=12*334661;333505), ... give the positive integer solutions to x^2 - 145*y^2 = -1.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (578,-1).

Cf. A097768 for S(n, 2*289).

Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.

I:=[1, 579]; [n le 2 select I[n] else 578*Self(n-1)-Self(n-2): n in [1..15]]; // Vincenzo Librandi, May 20 2012

LinearRecurrence[{578, -1}, {1, 579}, 20] (* or *) CoefficientList[Series[(1 + x)/(1 - 578 x + x^2), {x, 0, 20}], x] (* Harvey P. Dale, May 15 2011 *)

x='x+O('x^99); Vec((1+x)/(1-2*289*x+x^2)) \\ Altug Alkan, Apr 05 2018

G.f.: (1 + x)/(1 - 2*289*x + x^2).
a(n) = S(n, 2*289) + S(n-1, 2*289) = S(2*n, 2*sqrt(145)), with Chebyshev polynomials of the 2nd kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x).
a(n) = ((-1)^n)*T(2*n+1, 12*i)/(12*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.
a(n) = 578*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=579. - Philippe Deléham, Nov 18 2008
a(n) = (1/12)*sinh((2*n + 1)*arcsinh(12)). - Bruno Berselli, Apr 05 2018

A097772 Pell equation solutions (13a(n))^2 - 170b(n)^2 = -1 with b(n):=A097771(n), n >= 0.

Original entry on oeis.org

1, 679, 460361, 312124079, 211619665201, 143477820882199, 97277750938465721, 65954171658458876639, 44716831106684179895521, 30317945536160215510286599, 20555522356685519431794418601, 13936613839887246014541105524879

Offset: 0

Wolfdieter Lang, Aug 31 2004

			(x,y) = (13*1=13;1), (8827=13*679;677), (5984693=13*460361;459005), ... give the positive integer solutions to x^2 - 170*y^2 =-1.

Michael De Vlieger, Table of n, a(n) for n = 0..353
Tanya Khovanova, Recursive Sequences
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (678,-1).

Cf. A097771 for S(n, 2*339).

Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.

LinearRecurrence[{678, -1}, {1, 679}, 12] (* Ray Chandler, Aug 12 2015 *)

x='x+O('x^99); Vec((1+x)/(1-2*339*x+x^2)) \\ Altug Alkan, Apr 05 2018

G.f.: (1 + x)/(1 - 2*339*x + x^2).
a(n) = S(n, 2*339) + S(n-1, 2*339) = S(2*n, 2*sqrt(170)), with Chebyshev polynomials of the 2nd kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x).
a(n) = ((-1)^n)*T(2*n+1, 13*i)/(13*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.
a(n) = 678*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=679. - Philippe Deléham, Nov 18 2008
a(n) = (1/13)*sinh((2*n + 1)*arcsinh(13)). - Bruno Berselli, Apr 05 2018

A097774 Chebyshev U(n,x) polynomial evaluated at x=393=2*14^2+1.

Original entry on oeis.org

1, 786, 617795, 485586084, 381670044229, 299992169177910, 235793463303793031, 185333362164612144456, 145671786867921841749385, 114497839144824403002872154, 89995155896045112838415763659

Offset: 0

Wolfdieter Lang, Aug 31 2004

Used to form integer solutions of Pell equation a^2 - 197*b^2 =-1. See A097775 with A097776.

Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (786, -1).

LinearRecurrence[{786,-1},{1,786},30] (* or *) CoefficientList[ Series[ 1/(1-786x+x^2), {x,0,30}],x] (* Harvey P. Dale, Jun 15 2011 *)

a(n) = 2*393*a(n-1) - a(n-2), n>=1, a(0)=1, a(-1):=0.
a(n) = S(n, 2*393)= U(n, 393), Chebyshev's polynomials of the second kind. See A049310.
G.f.: 1/(1-2*393*x+x^2).
a(n)= sum((-1)^k*binomial(n-k, k)*786^(n-2*k), k=0..floor(n/2)), n>=0.
a(n) = ((393+28*sqrt(197))^(n+1) - (393-28*sqrt(197))^(n+1))/(56*sqrt(197)), n>=0.

cp's OEIS Frontend

A097741 Pell equation solutions (10*a(n))^2 - 101*b(n)^2 = -1 with b(n) = A097742(n), n >= 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A097766 Pell equation solutions (11*a(n))^2 - 122*b(n)^2 = -1 with b(n):=A097767(n), n >= 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A097769 Pell equation solutions (12*a(n))^2 - 145*b(n)^2 = -1 with b(n):=A097770(n), n >= 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A097772 Pell equation solutions (13*a(n))^2 - 170*b(n)^2 = -1 with b(n):=A097771(n), n >= 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A097774 Chebyshev U(n,x) polynomial evaluated at x=393=2*14^2+1.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

Formula

A097741 Pell equation solutions (10a(n))^2 - 101b(n)^2 = -1 with b(n) = A097742(n), n >= 0.

A097766 Pell equation solutions (11a(n))^2 - 122b(n)^2 = -1 with b(n):=A097767(n), n >= 0.

A097769 Pell equation solutions (12a(n))^2 - 145b(n)^2 = -1 with b(n):=A097770(n), n >= 0.

A097772 Pell equation solutions (13a(n))^2 - 170b(n)^2 = -1 with b(n):=A097771(n), n >= 0.