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 A001085 a(n) = 20*a(n-1) - a(n-2). (Formerly M4744 N2030) 11
 1, 10, 199, 3970, 79201, 1580050, 31521799, 628855930, 12545596801, 250283080090, 4993116004999, 99612037019890, 1987247624392801, 39645340450836130, 790919561392329799, 15778745887395759850, 314783998186522867201, 6279901217843061584170 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS From Wolfdieter Lang, Nov 08 2002: (Start) Chebyshev's polynomials T(n,x) evaluated at x=10. 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) 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 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. Except for the first term, positive values of x (or y) satisfying x^2 - 20xy + y^2 + 99 = 0. - Colin Barker, Feb 18 2014 REFERENCES 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 "Questions D'Arithmétique", Question 3686, Solution by H.L. Mennessier, Mathesis, 65(4, Supplement) 1956, pp. 1-12. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 0..200 H. Brocard, Notes élémentaires sur le problème de Peel, Nouvelle Correspondance Mathématique, 4 (1878), 161-169. Tanya Khovanova, Recursive Sequences Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, and Fausto Jarquín-Zárate, The sequences of Fibonacci and Lucas for each real quadratic fields Q(Sqrt(d)), arXiv:1904.13002 [math.NT], 2019. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009. Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992 Index entries for sequences related to Chebyshev polynomials. Index entries for linear recurrences with constant coefficients, signature (20,-1). FORMULA 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 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). G.f.: (1-10*x)/(1-20*x+x^2). - Simon Plouffe in his 1992 dissertation a(n) = (((10+3*sqrt(11))^n + (10-3*sqrt(11))^n))/2. a(n) = sqrt(99*A075843(n)^2 + 1), (cf. Richardson comment). 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 EXAMPLE G.f. = 1 + 10*x + 199*x^2 + 3970*x^3 + 79201*x^4 + 1580050*x^5 + 31521799*x^6 + ... MATHEMATICA LinearRecurrence[{20, -1}, {1, 10}, 30] (* T. D. Noe, Dec 19 2011 *) a[ n_] := ChebyshevT[ n, 10]; (* Michael Somos, May 27 2014 *) a[ n_] := ((10 + Sqrt[99])^n + (10 - Sqrt[99])^n) / 2 // Simplify; (* Michael Somos, May 27 2014 *) a[ n_] := With[{m = Abs @ n}, SeriesCoefficient[ (1 - 10 x) / (1 - 20 x + x^2), {x, 0, m}]]; (* Michael Somos, May 27 2014 *) Table[LucasL[n, 20*I]*(-I)^n/2, {n, 0, 30}] (* G. C. Greubel, Jun 06 2019 *) PROG (Sage) [lucas_number2(n, 20, 1)/2 for n in range(0, 20)] # Zerinvary Lajos, Jun 27 2008 (PARI) {a(n) = n=abs(n); polsym( 1 - 20*x + x^2, n) [n+1] / 2}; /* Michael Somos, May 27 2014 */ (PARI) my(x='x+O('x^30)); Vec((1-10*x)/(1-20*x+x^2)) \\ G. C. Greubel, Dec 20 2017 (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 (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 CROSSREFS Cf. A090728, A001084. Sequence in context: A097127 A249846 A211419 * A079436 A285021 A354410 Adjacent sequences: A001082 A001083 A001084 * A001086 A001087 A001088 KEYWORD nonn,easy AUTHOR N. J. A. Sloane STATUS approved

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Last modified September 25 19:30 EDT 2023. Contains 365648 sequences. (Running on oeis4.)