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 A078363 A Chebyshev T-sequence with Diophantine property. 4
 2, 13, 167, 2158, 27887, 360373, 4656962, 60180133, 777684767, 10049721838, 129868699127, 1678243366813, 21687295069442, 280256592535933, 3621648407897687, 46801172710133998, 604793596823844287, 7815515585999841733, 100996909021174098242 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS a(n) gives the general (positive integer) solution of the Pell equation a^2 - 165*b^2 = +4 with companion sequence b(n)=A078362(n-1), n>=1. Except for the first term, positive values of x (or y) satisfying x^2 - 13xy + y^2 + 165 = 0. - Colin Barker, Feb 26 2014 REFERENCES O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108). LINKS Table of n, a(n) for n=0..18. Tanya Khovanova, Recursive Sequences Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2) Index entries for sequences related to Chebyshev polynomials. Index entries for linear recurrences with constant coefficients, signature (13,-1). FORMULA a(n) = 13*a(n-1)-a(n-2), n >= 1; a(-1)=13, a(0)=2. a(n) = S(n, 13) - S(n-2, 13) = 2*T(n, 13/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 13)=A078362(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120. G.f.: (2-13*x)/(1-13*x+x^2). a(n) = ap^n + am^n, with ap := (13+sqrt(165))/2 and am := (13-sqrt(165))/2. a(n) = sqrt(4 + 165*A078362(n-1)^2), n>=1, (Pell equation d=165, +4). E.g.f.: 2*exp(13*x/2)*cosh(sqrt(165)*x/2). - Stefano Spezia, Sep 24 2022 MATHEMATICA a[0] = 2; a[1] = 13; a[n_] := 13a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 16}] (* Robert G. Wilson v, Jan 30 2004 *) LinearRecurrence[{13, -1}, {2, 13}, 20] (* Harvey P. Dale, Oct 28 2016 *) PROG (PARI) a(n)=if(n<0, 0, 2*subst(poltchebi(n), x, 13/2)) (PARI) a(n)=if(n<0, 0, polsym(1-13*x+x^2, n)[n+1]) (PARI) Vec((2-13*x)/(1-13*x+x^2) + O(x^100)) \\ Colin Barker, Feb 26 2014 (Sage) [lucas_number2(n, 13, 1) for n in range(0, 20)] - Zerinvary Lajos, Jun 25 2008 CROSSREFS Cf. A078362. Cf. A077428, A078355 (Pell +4 equations). Cf. A049310, A053120. Sequence in context: A177448 A258224 A351021 * A143851 A088316 A006905 Adjacent sequences: A078360 A078361 A078362 * A078364 A078365 A078366 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Nov 29 2002 EXTENSIONS More terms from Colin Barker, Feb 26 2014 STATUS approved

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Last modified February 28 04:16 EST 2024. Contains 370379 sequences. (Running on oeis4.)