A001085 a(n) = 20*a(n-1) - a(n-2).
1, 10, 199, 3970, 79201, 1580050, 31521799, 628855930, 12545596801, 250283080090, 4993116004999, 99612037019890, 1987247624392801, 39645340450836130, 790919561392329799, 15778745887395759850, 314783998186522867201, 6279901217843061584170
Offset: 0
Examples
G.f. = 1 + 10*x + 199*x^2 + 3970*x^3 + 79201*x^4 + 1580050*x^5 + 31521799*x^6 + ...
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).
Programs
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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
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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
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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 *) -
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 -
Sage
[lucas_number2(n,20,1)/2 for n in range(0,20)] # Zerinvary Lajos, Jun 27 2008
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
Comments