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 A097308 Chebyshev T-polynomials T(n,13) with Diophantine property. 5
 1, 13, 337, 8749, 227137, 5896813, 153090001, 3974443213, 103182433537, 2678768828749, 69544807113937, 1805486216133613, 46873096812360001, 1216895030905226413, 31592397706723526737, 820185445343906468749, 21293229181234844660737, 552803773266762054710413 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n)^2 - 42 (2*b(n))^2 = +1 with b(n):=A097309(n) gives all nonnegative solutions of this D:= 42*4= 168 Pell equation. Numbers n such that 42*(n^2-1) is a square. [From Vincenzo Librandi, Nov 17 2010] Except for the first term, positive values of x (or y) satisfying x^2 - 26xy + y^2 + 168 = 0. - Colin Barker, Feb 20 2014 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (26,-1). FORMULA a(n)=26*a(n-1) - a(n-2), a(-1) := 13, a(0)=1. a(n)= T(n, 13)= (S(n, 26)-S(n-2, 26))/2 = S(n, 26)-13*S(n-1, 26) with T(n, x), resp. S(n, x), Chebyshev's polynomials of the first, resp.second, kind. See A053120 and A049310. S(n, 26)=A097309(n). a(n)= (ap^n + am^n)/2 with ap := 13+2*sqrt(42) and am := 13-2*sqrt(42). a(n)= sum(((-1)^k)*(n/(2*(n-k)))*binomial(n-k, k)*(2*13)^(n-2*k), k=0..floor(n/2)), n>=1. G.f.: (1-13*x)/(1-26*x+x^2). a(n)=sqrt(1 + 168*A097309(n)^2), n>=0. a(n) = Cosh[2n*ArcSinh[Sqrt[6]]] - Herbert Kociemba, Apr 24 2008 MATHEMATICA CoefficientList[Series[(1 - 13 x)/(1 - 26 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 22 2014 *) PROG (PARI) Vec((1-13*x)/(1-26*x+x^2) + O(x^100)) \\ Colin Barker, Feb 20 2014 (MAGMA) I:=[1, 13]; [n le 2 select I[n] else 26*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 22 2014 CROSSREFS Cf. A097309. Sequence in context: A258297 A266902 A029807 * A204195 A238652 A041315 Adjacent sequences:  A097305 A097306 A097307 * A097309 A097310 A097311 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Aug 31 2004 EXTENSIONS Additional terms from Colin Barker, Feb 20 2014 STATUS approved

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