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A077412 Chebyshev U(n,x) polynomial evaluated at x=8.

%I A077412 #67 Aug 22 2024 02:33:42
%S A077412 1,16,255,4064,64769,1032240,16451071,262184896,4178507265,
%T A077412 66593931344,1061324394239,16914596376480,269572217629441,
%U A077412 4296240885694576,68470281953483775,1091228270370045824,17391182043967249409
%N A077412 Chebyshev U(n,x) polynomial evaluated at x=8.
%C A077412 For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 16'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 A077412 For n>=2, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,15}. - _Milan Janjic_, Jan 23 2015
%H A077412 Vincenzo Librandi, <a href="/A077412/b077412.txt">Table of n, a(n) for n = 0..800</a>
%H A077412 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A077412 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 A077412 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A077412 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (16,-1).
%F A077412 a(n) = 16*a(n-1) - a(n-2), n>=1, a(-1)=0, a(0)=1.
%F A077412 a(n) = S(n, 16) with S(n, x) := U(n, x/2), Chebyshev's polynomials of the second kind. See A049310.
%F A077412 G.f.: 1/(1 - 16*x + x^2).
%F A077412 a(n) = (((8 + 3*sqrt(7))^(n+1) - (8 - 3*sqrt(7))^(n+1)))/(6*sqrt(7)).
%F A077412 a(n) = sqrt((A001081(n+1)^2-1)/63).
%F A077412 a(n) = Sum_{k=0..n} A101950(n,k)*15^k. - _Philippe Deléham_, Feb 10 2012
%F A077412 Product {n >= 0} (1 + 1/a(n)) = 1/7*(7 + 3*sqrt(7)). - _Peter Bala_, Dec 23 2012
%F A077412 Product {n >= 1} (1 - 1/a(n)) = 1/16*(7 + 3*sqrt(7)). - _Peter Bala_, Dec 23 2012
%p A077412 seq( simplify(ChebyshevU(n, 8)), n=0..20); # _G. C. Greubel_, Dec 22 2019
%t A077412 Table[GegenbauerC[n, 1, 8], {n, 0, 20}] (* _Vladimir Joseph Stephan Orlovsky_, Sep 11 2008 *)
%t A077412 CoefficientList[Series[1/(1-16x+x^2), {x,0,20}], x] (* _Vincenzo Librandi_, Dec 24 2012 *)
%t A077412 LinearRecurrence[{16,-1}, {1,16}, 30] (* _G. C. Greubel_, Jan 18 2018 *)
%t A077412 ChebyshevU[Range[21] -1, 8] (* _G. C. Greubel_, Dec 22 2019 *)
%o A077412 (Sage) [lucas_number1(n,16,1) for n in range(1,20)] # _Zerinvary Lajos_, Jun 25 2008
%o A077412 (Sage) [chebyshev_U(n,8) for n in (0..20)] # _G. C. Greubel_, Dec 22 2019
%o A077412 (Magma) I:=[1, 16, 255]; [n le 3 select I[n] else 16*Self(n-1)-Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Dec 24 2012
%o A077412 (PARI) vector( 21, n, polchebyshev(n-1, 2, 8) ) \\ _G. C. Greubel_, Jan 18 2018
%o A077412 (GAP) m:=8;; 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
%Y A077412 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), this sequence (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), A097316 (m=33).
%Y A077412 Cf. A323182.
%K A077412 nonn,easy
%O A077412 0,2
%A A077412 _Wolfdieter Lang_, Nov 08 2002