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A097769 Pell equation solutions (12*a(n))^2 - 145*b(n)^2 = -1 with b(n):=A097770(n), n>=0. 3
1, 579, 334661, 193433479, 111804216201, 64622643530699, 37351776156527821, 21589261995829549839, 12478556081813323279121, 7212583826026105025782099, 4168860972887006891578774101, 2409594429744863957227505648279 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (578,-1).

FORMULA

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.

G.f.: (1+x)/(1-2*289*x+x^2).

a(n)=578*a(n-1)-a(n-2), n>1 ; a(0)=1, a(1)=579 . [From Philippe Deléham, Nov 18 2008]

EXAMPLE

(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.

MATHEMATICA

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

PROG

(MAGMA) 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

CROSSREFS

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

Sequence in context: A073735 A250727 A252985 * A186786 A185609 A127694

Adjacent sequences:  A097766 A097767 A097768 * A097770 A097771 A097772

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Aug 31 2004

STATUS

approved

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Last modified April 30 08:33 EDT 2017. Contains 285645 sequences.