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 A029547 Expansion of 1/(1-34*x+x^2). 14
 1, 34, 1155, 39236, 1332869, 45278310, 1538129671, 52251130504, 1775000307465, 60297759323306, 2048348816684939, 69583562007964620, 2363792759454112141, 80299370259431848174, 2727814796061228725775 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Chebyshev sequence U(n,17)=S(n,34) with Diophantine property. b(n)^2 - 2*(12*a(n))^2 = 1 with the companion sequence b(n)=A056771(n+1). - Wolfdieter Lang, Dec 11 2002 More generally, for t(m)=m+sqrt(m^2-1) and u(n)=(t(m)^(n+1)-1/t(m)^(n+1))/(t(m)-1/t(m)), we can verify that ((u(n+1)-u(n-1))/2)^2-(m^2-1)*u(n)^2=1. - Bruno Berselli, Nov 21 2011 a(n) equals the number of 01-avoiding words of length n on alphabet {0,1,...,33}. - Milan Janjic, Jan 26 2015 LINKS Indranil Ghosh, Table of n, a(n) for n = 0..651 (terms 0..200 from Vincenzo Librandi) R. Flórez, R. A. Higuita, A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014). Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (34,-1). FORMULA a(n) = 34*a(n-1) - a(n-2), a(-1)=0, a(0)=1. a(n) = S(n, 34) with S(n, x):= U(n, x/2) Chebyshev's polynomials of the 2nd kind. See A049310. - Wolfdieter Lang, Dec 11 2002 a(n) = (ap^(n+1) - am^(n+1))/(ap - am) with ap = 17+12*sqrt(2) and am = 17-12*sqrt(2). a(n) = sum((-1)^k*binomial(n-k, k)*34^(n-2*k), k = 0..floor(n/2)). a(n) = sqrt((A056771(n+1)^2 -1)/2)/12. a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3) with a(-1)=0, a(0)=1, a(1)=34. Also a(n) = (sqrt(2)/48)*((17+12*sqrt(2))^n-(17-12*sqrt(2))^n) = (sqrt(2)/48)*((3+2*sqrt(2))^(2n+2)-(3-2*sqrt(2))^(2n+2)) = (sqrt(2)/48)*((1+sqrt(2))^(4n+4)-(1-sqrt(2))^(4n+4)). - Antonio Alberto Olivares, Mar 19 2008 a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*33^k. - Philippe Deléham, Feb 10 2012 Product {n >= 0} (1 + 1/a(n)) = 1/4*(4 + 3*sqrt(2)). - Peter Bala, Dec 23 2012 Product {n >= 1} (1 - 1/a(n)) = 2/17*(4 + 3*sqrt(2)). - Peter Bala, Dec 23 2012 MAPLE with (combinat):seq(fibonacci(4*n, 2)/12, n=1..15); # Zerinvary Lajos, Apr 21 2008 MATHEMATICA lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 17]], {n, 0, 8^2}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *) LinearRecurrence[{34, -1}, {1, 34}, 20] (* Vincenzo Librandi, Nov 22 2011 *) PROG (PARI) A029547(n, x=[0, 1], A=[17, 72*4; 1, 17]) = vector(n, i, (x*=A)[1]) \\ M. F. Hasler, May 26 2007 (Sage) [lucas_number1(n, 34, 1) for n in xrange(1, 16)] # Zerinvary Lajos, Nov 07 2009 (MAGMA) I:=[1, 34]; [n le 2 select I[n] else 34*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 22 2011 CROSSREFS A091761 is an essentially identical sequence. Cf. A200441, A200724. Sequence in context: A248163 A158696 * A091761 A264134 A264019 A009978 Adjacent sequences:  A029544 A029545 A029546 * A029548 A029549 A029550 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified October 19 05:08 EDT 2018. Contains 316336 sequences. (Running on oeis4.)