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%I #56 Sep 08 2022 08:45:14
%S 0,1,26,675,17524,454949,11811150,306634951,7960697576,206671502025,
%T 5365498355074,139296285729899,3616337930622300,93885489910449901,
%U 2437406399741075126,63278680903357503375,1642808297087554012624
%N Chebyshev polynomials of the second kind, U(n,x), evaluated at x=13.
%C b(n)^2 - 42*(2*a(n))^2 = +1 with b(n):=A097308(n) gives all nonnegative integer solutions of this D:=42*4=168 Pell equation.
%C For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 26'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,...,25}. - _Milan Janjic_, Jan 25 2015
%H Indranil Ghosh, <a href="/A097309/b097309.txt">Table of n, a(n) for n = -1..705</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla and Fausto Jarquín-Zárate, <a href="https://arxiv.org/abs/1904.13002">The sequences of Fibonacci and Lucas for each real quadratic fields Q(Sqrt(d))</a>, arXiv:1904.13002 [math.NT], 2019.
%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 (26,-1).
%F a(n) = S(n, 26) = U(n, 13), n >= -1, with Chebyshev polynomials of 2nd kind. See A049310 for the triangle of S(n, x) coefficients. S(-1, x) := 0 =: U(-1, x).
%F a(n) = ((13+2*sqrt(42))^n - (13-2*sqrt(42))^n)/(4*sqrt(42)), (Binet form).
%F a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n-k, k) * 26^(n-2*k).
%F G.f.: 1/(1-26*x+x^2).
%F a(n) = 26*a(n-1) - a(n-2), a(-1)=0, a(0)=1. - _Philippe Deléham_, Nov 18 2008
%F a(n) = Sum_{k=0..n} A101950(n,k)*25^k. - _Philippe Deléham_, Feb 10 2012
%F With an offset of 0, product {n >= 1} (1 + 1/a(n)) = 1/6*(6 + sqrt(42)). - _Peter Bala_, Dec 23 2012
%F Product {n >= 2} (1 - 1/a(n)) = 1/13*(6 + sqrt(42)). - _Peter Bala_, Dec 23 2012
%F a(n) = sqrt((A097308(n)^2 - 1)/168).
%p seq( simplify(ChebyshevU(n, 13)), n=-1..20); # _G. C. Greubel_, Dec 22 2019
%t Table[GegenbauerC[n,1,13], {n,-1,20}] (* _Vladimir Joseph Stephan Orlovsky_, Sep 11 2008 *)
%t ChebyshevU[Range[22] -2, 13] (* _G. C. Greubel_, Dec 22 2019 *)
%o (Sage) [lucas_number1(n,26,1) for n in range(0,20)] # _Zerinvary Lajos_, Jun 25 2008
%o (Sage) [chebyshev_U(n,13) for n in (-1..20)] # _G. C. Greubel_, Dec 22 2019
%o (PARI) vector( 22, n, polchebyshev(n-2, 2, 13) ) \\ _G. C. Greubel_, Dec 22 2019
%o (Magma) m:=13; 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 22 2019
%o (GAP) m:=13;; 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 22 2019
%Y Cf. A097308, A101950.
%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), this sequence (m=13), A097311 (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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