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 A078987 Chebyshev U(n,x) polynomial evaluated at x=19. 27
 1, 38, 1443, 54796, 2080805, 79015794, 3000519367, 113940720152, 4326746846409, 164302439443390, 6239165952002411, 236924003736648228, 8996872976040630253, 341644249085807301386, 12973484592284636822415, 492650770257730391950384, 18707755785201470257292177 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS A078986(n+1)^2 - 10*(6*a(n))^2 = +1, n>=0 (Pell equation +1, see A033313 and A033317). a(n) equals the number of 01-avoiding words of length n on alphabet {0,1,...,37}. - Milan Janjic, Jan 26 2015 LINKS Colin Barker, Table of n, a(n) for n = 0..632 Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5. R. Flórez, R. A. Higuita, and 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 (38,-1). FORMULA a(n) = 38*a(n-1) - a(n-2), n>=1, a(-1)=0, a(0)=1. a(n) = S(n, 38) with S(n, x) = U(n, x/2), Chebyshev's polynomials of the second kind. See A049310. G.f.: 1/(1-38*x+x^2). a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*38^(n-2*k). a(n) = ((19+6*sqrt(10))^(n+1) - (19-6*sqrt(10))^(n+1))/(12*sqrt(10)). a(n) = Sum_{k=0..n} A101950(n,k)*37^k. - Philippe Deléham, Feb 10 2012 Product_{n>=0} (1 + 1/a(n)) = 1/3*(3 + sqrt(10)). - Peter Bala, Dec 23 2012 Product_{n>=1} (1 - 1/a(n)) = 3/19*(3 + sqrt(10)). - Peter Bala, Dec 23 2012 MAPLE seq( simplify(ChebyshevU(n, 19)), n=0..20); # G. C. Greubel, Dec 22 2019 MATHEMATICA lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 19]], {n, 0, 8^2}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *) ChebyshevU[Range[21] -1, 19] (* G. C. Greubel, Dec 22 2019 *) PROG (Sage) [lucas_number1(n, 38, 1) for n in range(1, 16)] # Zerinvary Lajos, Nov 07 2009 (Sage) [chebyshev_U(n, 19) for n in (0..20)] # G. C. Greubel, Dec 22 2019 (PARI) a(n)=subst(polchebyshev(n, 2), x, 19) \\ Charles R Greathouse IV, Feb 10 2012 (PARI) Vec(1/(1-38*x+x^2) + O(x^50)) \\ Colin Barker, Jun 15 2015 (MAGMA) m:=19; I:=[1, 2*m]; [n le 2 select I[n] else 2*m*Self(n-1) -Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 22 2019 (GAP) m:=19;; a:=[1, 2*m];; for n in [3..20] do a[n]:=2*m*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 22 2019 CROSSREFS Chebyshev sequence U(n, m): A000027 (m=1), A001353 (m=2), A001109 (m=3), A001090 (m=4), A004189 (m=5), A004191 (m=6), A007655 (m=7), A077412 (m=8), A049660 (m=9), A075843 (m=10), A077421 (m=11), A077423 (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), this sequence (m=19), A097316 (m=33). Sequence in context: A218740 A158702 A239364 * A009982 A041685 A221385 Adjacent sequences:  A078984 A078985 A078986 * A078988 A078989 A078990 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Jan 10 2003 STATUS approved

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Last modified January 18 08:37 EST 2021. Contains 340250 sequences. (Running on oeis4.)