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 A144128 Chebyshev U(n,x) polynomial evaluated at x=18. 23
 1, 36, 1295, 46584, 1675729, 60279660, 2168392031, 78001833456, 2805897612385, 100934312212404, 3630829342034159, 130608922001017320, 4698290362694589361, 169007844135004199676, 6079584098497456598975 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS From Bruno Berselli, Nov 21 2011: (Start) A Diophantine property of these numbers: ((a(n+1)-a(n-1))/2)^2 - 323*a(n)^2 = 1. More generally, for t(m) = m + sqrt(m^2-1) and u(n) = (t(m)^n - 1/t(m)^n)/(t(m) - 1/t(m)), we can verify that ((u(n+1) - u(n-1))/2)^2 - (m^2-1)*u(n)^2 = 1. (End) a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,35}. - Milan Janjic, Jan 26 2015 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..200 Index entries for linear recurrences with constant coefficients, signature (36,-1). FORMULA From Bruno Berselli, Nov 21 2011: (Start) G.f.: x/(1-36*+x^2). a(n) = 36*a(n-1) - a(n-2) with a(1)=1, a(2)=36. a(n) = (t^n - 1/t^n)/(t - 1/t) for t = 18+sqrt(323). a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-1-k, k)*36^(n-1-2*k). (End) a(n) = Sum_{k=0..n} A101950(n,k)*35^k. - Philippe Deléham, Feb 10 2012 Product {n >= 1} (1 + 1/a(n)) = 1/17*(17 + sqrt(323)). - Peter Bala, Dec 23 2012 Product {n >= 2} (1 - 1/a(n)) = 1/36*(17 + sqrt(323)). - Peter Bala, Dec 23 2012 MAPLE seq( simplify(ChebyshevU(n, 18)), n=0..20); # G. C. Greubel, Dec 22 2019 MATHEMATICA LinearRecurrence[{36, -1}, {1, 36}, 20] (* Vincenzo Librandi, Nov 22 2011 *) GegenbauerC[Range[0, 20], 1, 18] (* Harvey P. Dale, May 19 2019 *) ChebyshevU[Range[21] -1, 18] (* G. C. Greubel, Dec 22 2019 *) PROG (Sage) [lucas_number1(n, 36, 1) for n in range(1, 16)] # Zerinvary Lajos, Nov 07 2009 (PARI) Vec(x/(1-36*x+x^2)+O(x^16)) \\ Bruno Berselli, Nov 21 2011 (PARI) a(n) = polchebyshev(n, 2, 18); \\ Michel Marcus, Feb 09 2018 (Magma) Z:=PolynomialRing(Integers()); N:=NumberField(x^2-323); S:=[((18+r)^n-1/(18+r)^n)/(2*r): n in [1..15]]; [Integers()!S[j]: j in [1..#S]]; // Bruno Berselli, Nov 21 2011 (Magma) I:=[1, 36]; [n le 2 select I[n] else 36*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 22 2011 (Maxima) makelist(sum((-1)^k*binomial(n-1-k, k)*36^(n-1-2*k), k, 0, floor(n/2)), n, 1, 15); \\ Bruno Berselli, Nov 21 2011 (GAP) a:=[1, 36];; for n in [3..20] do a[n]:=36*a[n-1]-a[n-2]; od; a; # Muniru A Asiru, Feb 09 2018 CROSSREFS Cf. A200441, A200442, A200724 (incomplete list). 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), this sequence (m=18), A078987 (m=19), A097316 (m=33). Sequence in context: A170755 A218738 A158700 * A223405 A224267 A223924 Adjacent sequences: A144125 A144126 A144127 * A144129 A144130 A144131 KEYWORD nonn,easy AUTHOR Vladimir Joseph Stephan Orlovsky, Sep 11 2008 EXTENSIONS As Michel Marcus points out, some parts of this entry assume the offset is 1, others parts assume the offset is 0. The whole entry needs careful editing. - N. J. A. Sloane, Feb 10 2018 STATUS approved

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Last modified February 6 13:20 EST 2023. Contains 360110 sequences. (Running on oeis4.)