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 A097772 Pell equation solutions (13*a(n))^2 - 170*b(n)^2 = -1 with b(n):=A097771(n), n>=0. 3
 1, 679, 460361, 312124079, 211619665201, 143477820882199, 97277750938465721, 65954171658458876639, 44716831106684179895521, 30317945536160215510286599, 20555522356685519431794418601, 13936613839887246014541105524879 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (678,-1). FORMULA G.f.: (1 + x)/(1 - 2*339*x + x^2). a(n) = S(n, 2*339) + S(n-1, 2*339) = S(2*n, 2*sqrt(170)), with Chebyshev polynomials of the 2nd kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x). a(n) = ((-1)^n)*T(2*n+1, 13*I)/(13*I) with the imaginary unit I and Chebyshev polynomials of the first kind. See the T-triangle A053120. a(n) = 678*a(n-1)-a(n-2), n>1; a(0)=1, a(1)=679. - Philippe Deléham, Nov 18 2008 a(n) = (1/13)*sinh((2*n + 1)*arcsinh(13)). - Bruno Berselli, Apr 05 2018 EXAMPLE (x,y) = (13*1=13;1), (8827=13*679;677), (5984693=13*460361;459005), ... give the positive integer solutions to x^2 - 170*y^2 =-1. MATHEMATICA LinearRecurrence[{678, -1}, {1, 679}, 12] (* Ray Chandler, Aug 12 2015 *) PROG (PARI) x='x+O('x^99); Vec((1+x)/(1-2*339*x+x^2)) \\ Altug Alkan, Apr 05 2018 CROSSREFS Cf. A097771 for S(n, 2*339). Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775. Sequence in context: A199995 A164650 A200828 * A257828 A253695 A253702 Adjacent sequences:  A097769 A097770 A097771 * A097773 A097774 A097775 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Aug 31 2004 STATUS approved

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Last modified December 9 16:41 EST 2018. Contains 318023 sequences. (Running on oeis4.)