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A227152
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Nonnegative solutions of the Pell equation x^2 - 101*y^2 = +1. Solutions x = a(n).
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0
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1, 201, 80801, 32481801, 13057603201, 5249124005001, 2110134792407201, 848268937423689801, 341002002709530892801, 137081956820293995216201, 55106605639755476546020001, 22152718385224881277504824201
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OFFSET
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0,2
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COMMENTS
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The Pell equation x^2 - 101*y^2 = +1 has only proper solutions, namely x(n) = a(n) and y(n) = 20*A097740(n), n>= 0.
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REFERENCES
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T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964, ch. VI, 56., pp. 115-200.
O. Perron, Die Lehre von den Kettenbruechen, Band I, Teubner, Stuttgart, 1954, Paragraph 27, pp. 92-95.
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LINKS
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FORMULA
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a(n) = (S(n, 2*201) - S(n-2, 2*201))/2 = T(n, 201) with the Chebyshev S- and T-polynomials (see A049310 and A053120, respectively). S(n, -2) = -1, S(n, -1) = 0. For S(n, 2*201) see A097740.
a(n) = 2*201*a(n-1) - a(n-2), n >= 1, with input a(-1) = 201 and a(0) = 1.
O.g.f.: (1 - 201*x)/(1 - 2*201*x + x^2).
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EXAMPLE
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n=0: 1^2 - 101*0^2 = +1 (a proper, but not a positive solution),
n=1: 201^2 - 101*(20*1)^2 = +1, where 20 is the positive fundamental y-solution.
n=2: 80801^2 - 101*(20*402)^2 = +1, where 80801 = 7^2*17*97 and 20*402 = 8040 = 2^3*3*5*67.
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MATHEMATICA
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LinearRecurrence[{402, -1}, {1, 201}, 20] (* Harvey P. Dale, Jan 17 2020 *)
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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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