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A227152 Nonnegative solutions of the Pell equation x^2 - 101*y^2 = +1. Solutions x = a(n). 0
1, 201, 80801, 32481801, 13057603201, 5249124005001, 2110134792407201, 848268937423689801, 341002002709530892801, 137081956820293995216201, 55106605639755476546020001, 22152718385224881277504824201 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

REFERENCES

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.

LINKS

Table of n, a(n) for n=0..11.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (402,-1).

FORMULA

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).

EXAMPLE

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.

MATHEMATICA

LinearRecurrence[{402, -1}, {1, 201}, 20] (* Harvey P. Dale, Jan 17 2020 *)

CROSSREFS

Cf.  A097740 (y/20 solutions and S(n,402)), A049310, A053120.

Sequence in context: A305724 A167070 A175188 * A210166 A216376 A166505

Adjacent sequences:  A227149 A227150 A227151 * A227153 A227154 A227155

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Jul 05 2013

STATUS

approved

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Last modified May 14 22:40 EDT 2021. Contains 343909 sequences. (Running on oeis4.)