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 A004191 Expansion of 1/(1 - 12*x + x^2). 16
 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 a(n) = -a(-2-n) for all n in Z. - Michael Somos, Jun 29 2019 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..900 Andersen, K., Carbone, L. and Penta, D., Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9. D. Birmajer, J. B. Gil, M. D. Weiner, n the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 12 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 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 *) a[ n_] := ChebyshevU[n, 6]; (* Michael Somos, Jun 29 2019 *) 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 (PARI) {a(n) = polchebyshev(n, 2, 6)}; (* Michael Somos, Jun 29 2019 *) CROSSREFS Cf. A077417. Sequence in context: A219307 A172210 A171317 * A051051 A328468 A208382 Adjacent sequences:  A004188 A004189 A004190 * A004192 A004193 A004194 KEYWORD nonn,easy AUTHOR EXTENSIONS Chebyshev comments and a(n) formulas from Wolfdieter Lang, Nov 08 2002 STATUS approved

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Last modified October 23 14:45 EDT 2019. Contains 328345 sequences. (Running on oeis4.)