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A004191 Expansion of 1/(1 - 12*x + x^2). 15
1, 12, 143, 1704, 20305, 241956, 2883167, 34356048, 409389409, 4878316860, 58130412911, 692686638072, 8254109243953, 98356624289364, 1172025382228415, 13965947962451616, 166419350167190977, 1983066254043840108, 23630375698358890319, 281581442126262843720 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Chebyshev's polynomials U(n,x) evaluated at x=6.

a(n) give all (nontrivial, integer) solutions of Pell equation b(n)^2 - 35*a(n)^2 = +1 with b(n)=A023038(n+1), n>=0.

For positive n, a(n) equals the permanent of the tridiagonal matrix of order n with 12's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011

For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,11}. - Milan Janjic, Jan 26 2015

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..900

M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, 2014; http://matinf.pmfbl.org/wp-content/uploads/2015/01/za-arhiv-18.-1.pdf

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

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

FORMULA

a(n) = S(n, 12) with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. See A049310.

a(n) = ((6+sqrt(35))^(n+1) - (6-sqrt(35))^(n+1))/(2*sqrt(35)).

a(n) = sqrt((A023038(n)^2 - 1)/35).

[A077417(n), a(n)] = the 2 X 2 matrix [1,10; 1,11]^(n+1) * [1,0]. - Gary W. Adamson, Mar 19 2008

a(n) = 12*a(n-1)-a(n-2)for n>1, a(0)=1, a(1)=12. - Philippe Deléham, Nov 17 2008

a(n) = b such that (-1)^(n+1)*Integral_{x=0..Pi/2} (sin((n+1)*x))/(6+cos(x)) dx = c + b*(log(2)+log(3)-log(7)). - Francesco Daddi, Aug 01 2011

a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*11^k. - Philippe Deléham, Feb 10 2012

Product {n >= 0} (1 + 1/a(n)) = 1/5*(5 + sqrt(35)). - Peter Bala, Dec 23 2012

Product {n >= 1} (1 - 1/a(n)) = 1/12*(5 + sqrt(35)). - Peter Bala, Dec 23 2012

EXAMPLE

G.f. = 1 + 12*x + 143*x^2 + 1704*x^3 + 20305*x^4 + 241956*x^5 + ...

MATHEMATICA

lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 6]], {n, 0, 8^2}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)

CoefficientList[Series[1/(1 - 12*x + x^2), {x, 0, 30}], x] (* T. D. Noe, Aug 01 2011 *)

LinearRecurrence[{12, -1}, {1, 12}, 30] (* Harvey P. Dale, Feb 17 2016 *)

PROG

(Sage) [lucas_number1(n, 12, 1) for n in xrange(1, 20)] # Zerinvary Lajos, Jun 25 2008

(MAGMA) I:=[1, 12]; [n le 2 select I[n] else 12*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Jun 13 2012

(PARI) Vec(1/(1-12*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

CROSSREFS

Cf. A077417.

Sequence in context: A219307 A172210 A171317 * A051051 A208382 A208070

Adjacent sequences:  A004188 A004189 A004190 * A004192 A004193 A004194

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Chebyshev comments and a(n) formulas from Wolfdieter Lang, Nov 08 2002

STATUS

approved

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Last modified March 26 06:49 EDT 2017. Contains 284111 sequences.