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A001085 a(n) = 20*a(n-1) - a(n-2).

%I A001085 M4744 N2030 #95 Sep 08 2022 08:44:28
%S A001085 1,10,199,3970,79201,1580050,31521799,628855930,12545596801,
%T A001085 250283080090,4993116004999,99612037019890,1987247624392801,
%U A001085 39645340450836130,790919561392329799,15778745887395759850,314783998186522867201,6279901217843061584170
%N A001085 a(n) = 20*a(n-1) - a(n-2).
%C A001085 From _Wolfdieter Lang_, Nov 08 2002: (Start)
%C A001085 Chebyshev's polynomials T(n,x) evaluated at x=10.
%C A001085 The a(n) give all (unsigned, integer) solutions of Pell equation a(n)^2 - 99*b(n)^2 = +1 with b(n) = A075843(n), n >= 0. (End)
%C A001085 a(11+22k) - 1 and a(11+22k) + 1 are consecutive odd powerful numbers. The first pair is 99612037019890 +- 1. See A076445. - _T. D. Noe_, May 04 2006
%C A001085 This sequence gives the values of x in solutions of the Diophantine equation x^2 - 11*y^2 = 1. The corresponding y values are in A001084.
%C A001085 Except for the first term, positive values of x (or y) satisfying x^2 - 20xy + y^2 + 99 = 0. - _Colin Barker_, Feb 18 2014
%D A001085 Bastida, Julio R. Quadratic properties of a linearly recurrent sequence. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 163--166, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561042 (81e:10009) - From _N. J. A. Sloane_, May 30 2012
%D A001085 "Questions D'Arithmétique", Question 3686, Solution by H.L. Mennessier, Mathesis, 65(4, Supplement) 1956, pp. 1-12.
%D A001085 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001085 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001085 T. D. Noe, <a href="/A001085/b001085.txt">Table of n, a(n) for n = 0..200</a>
%H A001085 H. Brocard, <a href="https://gdz.sub.uni-goettingen.de/id/PPN598948236_0004?tify={%22pages%22:[186],%22view%22:%22info%22}">Notes élémentaires sur le problème de Peel</a>, Nouvelle Correspondance Mathématique, 4 (1878), 161-169.
%H A001085 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A001085 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 A001085 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A001085 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A001085 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A001085 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (20,-1).
%F A001085 For all members x of the sequence, 11*x^2 - 11 is a square. Limit_{n->infinity} a(n)/a(n-1) = 10 + 3*sqrt(11). - _Gregory V. Richardson_, Oct 13 2002
%F A001085 a(n) = T(n, 10) = (S(n, 20)-S(n-2, 20))/2, with S(n, x) := U(n, x/2) and T(n), resp. U(n, x), are Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(-2, x) := -1, S(-1, x) := 0, S(n-1, 20)= A075843(n).
%F A001085 G.f.: (1-10*x)/(1-20*x+x^2). - _Simon Plouffe_ in his 1992 dissertation
%F A001085 a(n) = (((10+3*sqrt(11))^n + (10-3*sqrt(11))^n))/2.
%F A001085 a(n) = sqrt(99*A075843(n)^2 + 1), (cf. Richardson comment).
%F A001085 a(n) = (-i)^n*Lucas(n, 20*i)/2, where i = sqrt(-1) and Lucas(n, x) is the Lucas polynomial. - _G. C. Greubel_, Jun 06 2019
%e A001085 G.f. = 1 + 10*x + 199*x^2 + 3970*x^3 + 79201*x^4 + 1580050*x^5 + 31521799*x^6 + ...
%t A001085 LinearRecurrence[{20, -1}, {1, 10}, 30] (* _T. D. Noe_, Dec 19 2011 *)
%t A001085 a[ n_] := ChebyshevT[ n, 10]; (* _Michael Somos_, May 27 2014 *)
%t A001085 a[ n_] := ((10 + Sqrt[99])^n + (10 - Sqrt[99])^n) / 2 // Simplify; (* _Michael Somos_, May 27 2014 *)
%t A001085 a[ n_] := With[{m = Abs @ n}, SeriesCoefficient[ (1 - 10 x) / (1 - 20 x + x^2), {x, 0, m}]]; (* _Michael Somos_, May 27 2014 *)
%t A001085 Table[LucasL[n, 20*I]*(-I)^n/2, {n,0,30}] (* _G. C. Greubel_, Jun 06 2019 *)
%o A001085 (Sage) [lucas_number2(n,20,1)/2 for n in range(0,20)] # _Zerinvary Lajos_, Jun 27 2008
%o A001085 (PARI) {a(n) = n=abs(n); polsym( 1 - 20*x + x^2, n) [n+1] / 2}; /* _Michael Somos_, May 27 2014 */
%o A001085 (PARI) my(x='x+O('x^30)); Vec((1-10*x)/(1-20*x+x^2)) \\ _G. C. Greubel_, Dec 20 2017
%o A001085 (Magma) I:=[1,10]; [n le 2 select I[n] else 20*Self(n-1)-Self(n-2): n in [1..30]]; // _G. C. Greubel_, Dec 20 2017
%o A001085 (GAP) a:=[1,10];; for n in [3..30] do a[n]:=20*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Jun 06 2019
%Y A001085 Cf. A090728, A001084.
%K A001085 nonn,easy
%O A001085 0,2
%A A001085 _N. J. A. Sloane_