OFFSET
0,3
COMMENTS
The Pell equation x^2 - 89*y^2 = +1 has only proper solutions, namely x(n) = A227110(n) and y(n) = 2^3*5^3*53*a(n), n>= 0, where 2^3*5^3*53 = 53000 is the fundamental positive y solution.
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, p. 92-95.
LINKS
Index entries for linear recurrences with constant coefficients, signature (1000002, -1).
FORMULA
a(n) = S(n-1, 2*500001) with the Chebyshev S- polynomial (see A049310). S(n, -1) = 0, where 500001 = 3*166667 is the corresponding fundamental x-solution.
a(n) = 500001*a(n-1) - a(n-2), n >= 1 with inputs a(-1) = -1 and a(0) = 0.
O.g.f.: x/(1 - 2*500001*x + x^2).
MATHEMATICA
LinearRecurrence[{1000002, -1}, {0, 1}, 10] (* Ray Chandler, Aug 11 2015 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jul 02 2013
STATUS
approved