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A097783 Chebyshev polynomials S(n,11) + S(n-1,11) with Diophantine property. 9
1, 12, 131, 1429, 15588, 170039, 1854841, 20233212, 220710491, 2407582189, 26262693588, 286482047279, 3125039826481, 34088956044012, 371853476657651, 4056299287190149, 44247438682433988, 482665526219583719, 5265073349732986921, 57433141320843272412 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

All positive integer solutions of Pell equation (3*a(n))^2 - 13*b(n)^2 = -4 together with b(n)=A078922(n+1), n>=0.

a(n) = L(n,-11)*(-1)^n, where L is defined as in A108299; see also A078922 for L(n,+11). - Reinhard Zumkeller, Jun 01 2005

LINKS

Colin Barker, Table of n, a(n) for n = 0..963

S. Falcon, Relationships between Some k-Fibonacci Sequences, Applied Mathematics, 2014, 5, 2226-2234.

Tanya Khovanova, Recursive Sequences

Eric Weisstein's World of Mathematics, Fibonacci Polynomial

H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (11,-1).

FORMULA

a(n) = S(n, 11) + S(n-1, 11) = S(2*n, sqrt(13)), with S(n, x)=U(n, x/2) Chebyshev's polynomials of the 2nd kind, A049310. S(-1, x)= 0 = U(-1, x).

a(n) = (-2/3)*I*((-1)^n)*T(2*n+1, 3*I/2) with the imaginary unit I and Chebyshev's polynomials of the first kind. See the T-triangle A053120.

G.f.: (1+x)/(1-11*x+x^2).

a(n) = 11*a(n-1)-a(n-2) with a(0)=1 and a(1)=12. - Philippe Deléham, Nov 17 2008

From Peter Bala, Mar 22 2015: (Start)

The aerated sequence (b(n))n>=1 = [1, 0, 12, 0, 131, 0, 1429, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -9, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for the connection with Chebyshev polynomials.

b(n) = 1/2*( (-1)^n - 1 )*F(n,3) + 1/3*( 1 + (-1)^(n+1) )*F(n+1,3), where F(n,x) is the n-th Fibonacci polynomial. The o.g.f. is x*(1 + x^2)/(1 - 11*x^2 + x^4).

Exp( Sum_{n >= 1} 6*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 6*A006190(n)*x^n.

Exp( Sum_{n >= 1} (-6)*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 6*A006190(n)*(-x)^n. Cf. A002315, A004146, A113224 and A192425. (End)

EXAMPLE

All positive solutions to the Pell equation x^2 - 13*y^2 = -4 are (3=3*1,1), (36=3*12,10), (393=3*131,109), (4287=3*1429,1189 ), ...

MATHEMATICA

CoefficientList[Series[(1 + x) / (1 - 11 x + x^2), {x, 0, 33}], x] (* Vincenzo Librandi, Mar 22 2015 *)

PROG

(Sage) [(lucas_number2(n, 11, 1)-lucas_number2(n-1, 11, 1))/9 for n in xrange(1, 19)] # Zerinvary Lajos, Nov 10 2009

(PARI) Vec((1+x)/(1-11*x+x^2) + O(x^30)) \\ Michel Marcus, Mar 22 2015

(MAGMA) I:=[1, 12]; [n le 2 select I[n] else 11*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 22 2015

CROSSREFS

Cf. S(n, 11) = A004190(n).

Cf. A000045, A002315, A004146, A006190, A100047, A113224, A192425.

Sequence in context: A209013 A240798 A160962 * A260018 A078218 A048643

Adjacent sequences:  A097780 A097781 A097782 * A097784 A097785 A097786

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Aug 31 2004

STATUS

approved

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Last modified November 19 00:50 EST 2017. Contains 294912 sequences.