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A097311 Chebyshev polynomials of the second kind, U(n,x), evaluated at x=14. 3

%I

%S 0,1,28,783,21896,612305,17122644,478821727,13389885712,374437978209,

%T 10470873504140,292810020137711,8188209690351768,228977061309711793,

%U 6403169506981578436,179059769134174484415,5007270366249903985184

%N Chebyshev polynomials of the second kind, U(n,x), evaluated at x=14.

%C b(n)^2 - 195*a(n)^2 = +1 with b(n):=A097310(n) gives all nonnegative integer solutions of this Pell equation.

%C For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 28's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). [_John M. Campbell_, Jul 08 2011]

%C For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,27}. - _Milan Janjic_, Jan 26 2015

%H Indranil Ghosh, <a href="/A097311/b097311.txt">Table of n, a(n) for n = -1..689</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (28, -1)

%F a(n) = S(n, 28) = U(n, 14), n>=-1, with Chebyshev polynomials of the second kind. See A049310 for the triangle of S(n, x) coefficients. S(-1, x) := 0 =: U(-1, x).

%F G.f.: 1/(1-28*x+x^2).

%F a(n) = ((14+sqrt(195))^(n+1) - (14-sqrt(195))^(n+1))/(2*sqrt(195)), (Binet form).

%F a(n) = 28*a(n-1)-a(n-2); a(0)=1, a(1)=28; a(-1)=0. - _Zerinvary Lajos_, Apr 29 2009

%F a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*27^k. - _Philippe Deléham_, Feb 10 2012

%F With an offset of 0, product {n >= 1} (1 + 1/a(n)) = 1/13*(13 + sqrt(195)). - _Peter Bala_, Dec 23 2012

%F Product {n >= 2} (1 - 1/a(n)) = 1/28*(13 + sqrt(195)). - _Peter Bala_, Dec 23 2012

%F a(n) = sqrt((A097310(n)^2 - 1)/195).

%t lst={};Do[AppendTo[lst, GegenbauerC[n, 1, 14]], {n, 0, 8^2}];lst (* _Vladimir Joseph Stephan Orlovsky_, Sep 11 2008 *]

%t LinearRecurrence[{28, -1},{0, 1},17] (* _Ray Chandler_, Aug 12 2015 *)

%o (Sage) [lucas_number1(n,28,1) for n in xrange(0,20)] # _Zerinvary Lajos_, Jun 25 2008

%Y Cf. A049310, A097310.

%K nonn,easy

%O -1,3

%A _Wolfdieter Lang_, Aug 31 2004

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Last modified November 21 14:18 EST 2019. Contains 329371 sequences. (Running on oeis4.)