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A097729
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Pell equation solutions (6*a(n))^2 - 37*b(n)^2 = -1 with b(n):=A097730(n), n >= 0.
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4
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1, 147, 21461, 3133159, 457419753, 66780150779, 9749444593981, 1423352130570447, 207799661618691281, 30337327244198356579, 4429041977991341369253, 646609791459491641554359, 94400600511107788325567161, 13781841064830277603891251147, 2012054394864709422379797100301
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f.: (1 + x)/(1 - 2*73*x + x^2).
a(n) = S(n, 2*73) + S(n-1, 2*73) = S(2*n, 2*sqrt(37)), with Chebyshev polynomials of the second 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, 6*i)/(6*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.
a(n) = 146*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=147. - Philippe Deléham, Nov 18 2008
a(n) = (1/6)*sinh((2*n + 1)*arcsinh(6)). - Bruno Berselli, Apr 03 2018
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EXAMPLE
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(x,y) = (6,1), (882,145), (128766,21169), ... give the positive integer solutions to x^2 - 37*y^2 =-1.
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MATHEMATICA
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LinearRecurrence[{146, -1}, {1, 147}, 20] (* Harvey P. Dale, Sep 24 2012 *)
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PROG
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(PARI) x='x+O('x^99); Vec((1+x)/(1-2*73*x+x^2)) \\ Altug Alkan, Apr 05 2018
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CROSSREFS
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Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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