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 A097316 Chebyshev U(n,x) polynomial evaluated at x=33. 22
 1, 66, 4355, 287364, 18961669, 1251182790, 82559102471, 5447649580296, 359462313197065, 23719065021425994, 1565098829100918539, 103272803655639197580, 6814439942443086121741, 449649763397588044837326 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Used to form integer solutions of Pell equation a^2 - 17*b^2 =-1. See A078989 with A078988. LINKS Indranil Ghosh, Table of n, a(n) for n = 0..548 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 (66,-1). FORMULA a(n) = 66*a(n-1) - a(n-2), n>=1, a(0)=1, a(-1):=0. a(n) = S(n, 66) with S(n, x) := U(n, x/2), Chebyshev's polynomials of the second kind. See A049310. G.f.: 1/(1-66*x+x^2). a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*66^(n-2*k). a(n) = ((33+8*sqrt(17))^(n+1) - (33-8*sqrt(17))^(n+1))/(16*sqrt(17)). MAPLE seq( simplify(ChebyshevU(n, 33)), n=0..20); # G. C. Greubel, Dec 22 2019 MATHEMATICA LinearRecurrence[{66, -1}, {1, 66}, 14] (* Ray Chandler, Aug 11 2015 *) ChebyshevU[Range[21] -1, 33] (* G. C. Greubel, Dec 22 2019 *) PROG (PARI) vector( 21, n, polchebyshev(n-1, 2, 33) ) \\ G. C. Greubel, Dec 22 2019 (MAGMA) m:=33; 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 (Sage) [chebyshev_U(n, 33) for n in (0..20)] # G. C. Greubel, Dec 22 2019 (GAP) m:=33;; 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), A078987 (m=19), this sequence (m=33). Sequence in context: A004998 A239409 A295592 * A239337 A099639 A003555 Adjacent sequences:  A097313 A097314 A097315 * A097317 A097318 A097319 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Aug 31 2004 STATUS approved

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Last modified August 14 18:13 EDT 2022. Contains 356122 sequences. (Running on oeis4.)