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A097316
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Chebyshev U(n,x) polynomial evaluated at x=33.
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22
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1, 66, 4355, 287364, 18961669, 1251182790, 82559102471, 5447649580296, 359462313197065, 23719065021425994, 1565098829100918539, 103272803655639197580, 6814439942443086121741, 449649763397588044837326
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OFFSET
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0,2
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COMMENTS
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Used to form integer solutions of Pell equation a^2 - 17*b^2 =-1. See A078989 with A078988.
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LINKS
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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).
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FORMULA
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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)).
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MAPLE
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seq( simplify(ChebyshevU(n, 33)), n=0..20); # G. C. Greubel, Dec 22 2019
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MATHEMATICA
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LinearRecurrence[{66, -1}, {1, 66}, 14] (* Ray Chandler, Aug 11 2015 *)
ChebyshevU[Range[21] -1, 33] (* G. C. Greubel, Dec 22 2019 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang, Aug 31 2004
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STATUS
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approved
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