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%I #55 Sep 08 2022 08:45:38
%S 1,36,1295,46584,1675729,60279660,2168392031,78001833456,
%T 2805897612385,100934312212404,3630829342034159,130608922001017320,
%U 4698290362694589361,169007844135004199676,6079584098497456598975
%N Chebyshev U(n,x) polynomial evaluated at x=18.
%C From _Bruno Berselli_, Nov 21 2011: (Start)
%C A Diophantine property of these numbers: ((a(n+1)-a(n-1))/2)^2 - 323*a(n)^2 = 1.
%C More generally, for t(m) = m + sqrt(m^2-1) and u(n) = (t(m)^n - 1/t(m)^n)/(t(m) - 1/t(m)), we can verify that ((u(n+1) - u(n-1))/2)^2 - (m^2-1)*u(n)^2 = 1. (End)
%C a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,35}. - _Milan Janjic_, Jan 26 2015
%H Vincenzo Librandi, <a href="/A144128/b144128.txt">Table of n, a(n) for n = 1..200</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (36,-1).
%F From _Bruno Berselli_, Nov 21 2011: (Start)
%F G.f.: x/(1-36*+x^2).
%F a(n) = 36*a(n-1) - a(n-2) with a(1)=1, a(2)=36.
%F a(n) = (t^n - 1/t^n)/(t - 1/t) for t = 18+sqrt(323).
%F a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-1-k, k)*36^(n-1-2*k). (End)
%F a(n) = Sum_{k=0..n} A101950(n,k)*35^k. - _Philippe Deléham_, Feb 10 2012
%F Product {n >= 1} (1 + 1/a(n)) = 1/17*(17 + sqrt(323)). - _Peter Bala_, Dec 23 2012
%F Product {n >= 2} (1 - 1/a(n)) = 1/36*(17 + sqrt(323)). - _Peter Bala_, Dec 23 2012
%p seq( simplify(ChebyshevU(n, 18)), n=0..20); # _G. C. Greubel_, Dec 22 2019
%t LinearRecurrence[{36,-1},{1,36},20] (* _Vincenzo Librandi_, Nov 22 2011 *)
%t GegenbauerC[Range[0,20],1,18] (* _Harvey P. Dale_, May 19 2019 *)
%t ChebyshevU[Range[21] -1, 18] (* _G. C. Greubel_, Dec 22 2019 *)
%o (Sage) [lucas_number1(n,36,1) for n in range(1, 16)] # _Zerinvary Lajos_, Nov 07 2009
%o (PARI) Vec(x/(1-36*x+x^2)+O(x^16)) \\ _Bruno Berselli_, Nov 21 2011
%o (PARI) a(n) = polchebyshev(n, 2, 18); \\ _Michel Marcus_, Feb 09 2018
%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-323); S:=[((18+r)^n-1/(18+r)^n)/(2*r): n in [1..15]]; [Integers()!S[j]: j in [1..#S]]; // _Bruno Berselli_, Nov 21 2011
%o (Magma) I:=[1,36]; [n le 2 select I[n] else 36*Self(n-1)-Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Nov 22 2011
%o (Maxima) makelist(sum((-1)^k*binomial(n-1-k,k)*36^(n-1-2*k),k,0,floor(n/2)),n,1,15); \\ _Bruno Berselli_, Nov 21 2011
%o (GAP) a:=[1,36];; for n in [3..20] do a[n]:=36*a[n-1]-a[n-2]; od; a; # _Muniru A Asiru_, Feb 09 2018
%Y Cf. A200441, A200442, A200724 (incomplete list).
%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), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), this sequence (m=18), A078987 (m=19), A097316 (m=33).
%K nonn,easy
%O 1,2
%A _Vladimir Joseph Stephan Orlovsky_, Sep 11 2008
%E As _Michel Marcus_ points out, some parts of this entry assume the offset is 1, others parts assume the offset is 0. The whole entry needs careful editing. - _N. J. A. Sloane_, Feb 10 2018
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