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%I #43 Sep 08 2022 08:45:14
%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).
%p seq( simplify(ChebyshevU(n, 14)), n=-1..20); # _G. C. Greubel_, Dec 23 2019
%t Table[GegenbauerC[n, 1, 14], {n,0,20}] (* _Vladimir Joseph Stephan Orlovsky_, Sep 11 2008 *]
%t LinearRecurrence[{28, -1},{0, 1},17] (* _Ray Chandler_, Aug 12 2015 *)
%t ChebyshevU[Range[21] -2, 14] (* _G. C. Greubel_, Dec 23 2019 *)
%o (Sage) [lucas_number1(n,28,1) for n in range(0,20)] # _Zerinvary Lajos_, Jun 25 2008
%o (Sage) [chebyshev_U(n,14) for n in (-1..20)] # _G. C. Greubel_, Dec 23 2019
%o (PARI) vector( 22, n, polchebyshev(n-2, 2, 14) ) \\ _G. C. Greubel_, Dec 23 2019
%o (Magma) m:=14; I:=[0,1]; [n le 2 select I[n] else 2*m*Self(n-1) -Self(n-2): n in [1..20]]; // _G. C. Greubel_, Dec 23 2019
%o (GAP) m:=14;; a:=[0,1];; for n in [3..20] do a[n]:=2*m*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Dec 23 2019
%Y Cf. A049310, A097310.
%Y 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), this sequence (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), A097316 (m=33).
%K nonn,easy
%O -1,3
%A _Wolfdieter Lang_, Aug 31 2004
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