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A001085
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a(n) = 20*a(n-1) - a(n-2).
(Formerly M4744 N2030)
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11
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1, 10, 199, 3970, 79201, 1580050, 31521799, 628855930, 12545596801, 250283080090, 4993116004999, 99612037019890, 1987247624392801, 39645340450836130, 790919561392329799, 15778745887395759850, 314783998186522867201, 6279901217843061584170
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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FORMULA
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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
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EXAMPLE
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G.f. = 1 + 10*x + 199*x^2 + 3970*x^3 + 79201*x^4 + 1580050*x^5 + 31521799*x^6 + ...
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MATHEMATICA
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LinearRecurrence[{20, -1}, {1, 10}, 30] (* T. D. Noe, Dec 19 2011 *)
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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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