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%I A078986 #46 Dec 29 2020 20:39:42
%S A078986 1,19,721,27379,1039681,39480499,1499219281,56930852179,2161873163521,
%T A078986 82094249361619,3117419602578001,118379850648602419,
%U A078986 4495316905044313921,170703662541035326579,6482243859654298096081,246154563004322292324499,9347391150304592810234881,354954709148570204496600979,13478931556495363178060602321
%N A078986 Chebyshev T(n,19) polynomial.
%C A078986 a(n+1)^2 - 10*(6*A078987(n))^2 = 1, n >= 0 (Pell equation +1, see A033313 and A033317).
%C A078986 Also gives solutions to the equation x^2 - 1 = floor(x*r*floor(x/r)) where r=sqrt(10). - _Benoit Cloitre_, Feb 14 2004
%C A078986 Numbers n such that 10*(n^2 - 1) is a square. - _Vincenzo Librandi_, Aug 08 2010
%H A078986 Indranil Ghosh, <a href="/A078986/b078986.txt">Table of n, a(n) for n = 0..632</a>
%H A078986 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 A078986 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A078986 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A078986 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (38,-1).
%F A078986 a(n) = 38*a(n-1) - a(n-2), a(-1) := 19, a(0)=1.
%F A078986 G.f.: (1-19*x)/(1-38*x+x^2).
%F A078986 a(n) = T(n, 19) = (S(n, 38)-S(n-2, 38))/2 = S(n, 38)-19*S(n-1, 38) with T(n, x), resp. S(n, x), Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(n, 38) = A078987(n).
%F A078986 a(n) = (ap^n + am^n)/2 with ap := 19+6*sqrt(10) and am := 19-6*sqrt(10).
%F A078986 a(n) = Sum_{k=0..floor(n/2)} ((-1)^k)*(n/(2*(n-k)))*binomial(n-k, k)*(2*19)^(n-2*k), n >= 1.
%F A078986 a(n) = cosh(2*arcsinh(3)*n). - _Herbert Kociemba_, Apr 24 2008
%t A078986 LinearRecurrence[{38, -1},{1, 19},15] (* _Ray Chandler_, Aug 11 2015 *)
%o A078986 (Sage) [lucas_number2(n,38,1)/2 for n in range(0, 16)] # _Zerinvary Lajos_, Nov 07 2009
%o A078986 (PARI) a(n) = polchebyshev(n, 1, 19); \\ _Michel Marcus_, Jan 14 2018
%Y A078986 Row 3 of array A188645.
%K A078986 nonn,easy
%O A078986 0,2
%A A078986 _Wolfdieter Lang_, Jan 10 2003
%E A078986 More terms from _Indranil Ghosh_, Feb 04 2017

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