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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, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.

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 November 23 13:09 EST 2017. Contains 295127 sequences.