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%I #69 Jan 02 2023 21:56:50
%S 1,38,1443,54796,2080805,79015794,3000519367,113940720152,
%T 4326746846409,164302439443390,6239165952002411,236924003736648228,
%U 8996872976040630253,341644249085807301386,12973484592284636822415,492650770257730391950384,18707755785201470257292177
%N Chebyshev U(n,x) polynomial evaluated at x=19.
%C A078986(n+1)^2 - 10*(6*a(n))^2 = +1, n>=0 (Pell equation +1, see A033313 and A033317).
%C a(n) equals the number of 01-avoiding words of length n on alphabet {0,1,...,37}. - _Milan Janjic_, Jan 26 2015
%H Colin Barker, <a href="/A078987/b078987.txt">Table of n, a(n) for n = 0..632</a>
%H Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Nemeth/nemeth7.html">Ellipse Chains and Associated Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
%H R. Flórez, R. A. Higuita, and A. Mukherjee, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mukherjee/mukh2.html">Alternating Sums in the Hosoya Polynomial Triangle</a>, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
%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 (38,-1).
%F a(n) = 38*a(n-1) - a(n-2), n>=1, a(-1)=0, a(0)=1.
%F a(n) = S(n, 38) with S(n, x) = U(n, x/2), Chebyshev's polynomials of the second kind. See A049310.
%F G.f.: 1/(1-38*x+x^2).
%F a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*38^(n-2*k).
%F a(n) = ((19+6*sqrt(10))^(n+1) - (19-6*sqrt(10))^(n+1))/(12*sqrt(10)).
%F a(n) = Sum_{k=0..n} A101950(n,k)*37^k. - _Philippe Deléham_, Feb 10 2012
%F Product_{n>=0} (1 + 1/a(n)) = 1/3*(3 + sqrt(10)). - _Peter Bala_, Dec 23 2012
%F Product_{n>=1} (1 - 1/a(n)) = 3/19*(3 + sqrt(10)). - _Peter Bala_, Dec 23 2012
%F From _Andrea Pinos_, Jan 02 2023: (Start)
%F a(n) = (A097314(n+1) - A097315(n+1))/2.
%F a(n) = (A097314(n) + A097315(n))/2. (End)
%p seq( simplify(ChebyshevU(n, 19)), n=0..20); # _G. C. Greubel_, Dec 22 2019
%t lst={};Do[AppendTo[lst, GegenbauerC[n, 1, 19]], {n, 0, 8^2}];lst (* _Vladimir Joseph Stephan Orlovsky_, Sep 11 2008 *)
%t ChebyshevU[Range[21] -1, 19] (* _G. C. Greubel_, Dec 22 2019 *)
%o (Sage) [lucas_number1(n,38,1) for n in range(1, 16)] # _Zerinvary Lajos_, Nov 07 2009
%o (Sage) [chebyshev_U(n,19) for n in (0..20)] # _G. C. Greubel_, Dec 22 2019
%o (PARI) a(n)=subst(polchebyshev(n,2),x,19) \\ _Charles R Greathouse IV_, Feb 10 2012
%o (PARI) Vec(1/(1-38*x+x^2) + O(x^50)) \\ _Colin Barker_, Jun 15 2015
%o (Magma) m:=19; I:=[1, 2*m]; [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:=19;; 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 Cf. A097314, A097315.
%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), A144128 (m=18), this sequence (m=19), A097316 (m=33).
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Jan 10 2003
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