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A077417 Chebyshev T-sequence with Diophantine property. 16
1, 11, 131, 1561, 18601, 221651, 2641211, 31472881, 375033361, 4468927451, 53252096051, 634556225161, 7561422605881, 90102515045411, 1073668757939051, 12793922580223201, 152453402204739361, 1816646903876649131, 21647309444315050211 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

7*a(n)^2 - 5*b(n)^2 = 2 with companion sequence b(n)=A077416(n), n>=0.

a(n) = L(n,12), where L is defined as in A108299; see also A077416 for L(n,-12). - Reinhard Zumkeller, Jun 01 2005

[a(n), A004191(n)] = the 2 X 2 matrix [1,10; 1,11]^(n+1) * [1,0]. - Gary W. Adamson, Mar 19 2008

Hankel transform of A174227. [Paul Barry, Mar 12 2010]

Alternate denominators of the continued fraction convergents to sqrt(35), see A041059. [James R. Buddenhagen, May 20 2010]

For positive n, a(n) equals the permanent of the (2n)X(2n) tridiagonal matrix with sqrt(10)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. [John M. Campbell, Jul 08 2011]

Positive values of x (or y) satisfying x^2 - 12xy + y^2 + 10 = 0. - Colin Barker, Feb 09 2014

LINKS

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

Tanya Khovanova, Recursive Sequences

J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962, 2014

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (12,-1).

FORMULA

a(n) = 12*a(n-1) - a(n-2), a(-1)=1, a(0)=1.

a(n) = S(n, 12) - S(n-1, 12) = T(2*n+1, sqrt(14)/2)/(sqrt(14)/2) with S(n, x) := U(n, x/2), resp. T(n, x), Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120. S(-1, x)=0, S(n, 12)=A004191(n).

G.f.:(1-x)/(1-12*x+x^2).

a(n) = (ap^(2*n+1) + am^(2*n+1))/sqrt(14) with ap := (sqrt(7)+sqrt(5))/sqrt(2) and am := (sqrt(7)-sqrt(5))/sqrt(2).

a(n) = sqrt((5*A077416(n)^2 + 2)/7).

a(n)*a(n+3) = 120 + a(n+1)*a(n+2). - Ralf Stephan, May 29 2004

MATHEMATICA

CoefficientList[Series[(1 - x)/(1 - 12 x + x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 10 2014 *)

LinearRecurrence[{12, -1}, {1, 11}, 30] (* Harvey P. Dale, Apr 09 2015 *)

PROG

(MAGMA) I:=[1, 11]; [n le 2 select I[n] else 12*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 10 2014

CROSSREFS

Cf. A072256(n) with companion A054320(n-1), n>=1.

Row 12 of array A094954.

Cf. A004191.

Cf. A041059. [James R. Buddenhagen, May 20 2010]

Cf. similar sequences listed in A238379.

Sequence in context: A076255 A076357 A015606 * A082148 A075509 A061113

Adjacent sequences:  A077414 A077415 A077416 * A077418 A077419 A077420

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Nov 29 2002

EXTENSIONS

More terms from Vincenzo Librandi, Feb 10 2014

STATUS

approved

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Last modified November 19 07:02 EST 2017. Contains 294915 sequences.