OFFSET
1,1
COMMENTS
Lim. n-> Inf. a(n)/a(n-1) = 3 + 2*sqrt(2).
Positive values of x (or y) satisfying x^2 - 6*x*y + y^2 + 36 = 0. - Colin Barker, Feb 08 2014
For each member t of the sequence there exists a nonnegative r such that t^2 = r^2 + (r+3)^2. The r values are in A241976. Example: 87^2 = 60^2 + 63^2. - Bruno Berselli, Jul 10 2017
REFERENCES
A. H. Beiler, "The Pellian", ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966.
L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400.
Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, pp. 139-147.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..200
Tanya Khovanova, Recursive Sequences
J. J. O'Connor and E. F. Robertson, Pell's Equation
Eric Weisstein's World of Mathematics, Pell Equation.
Index entries for linear recurrences with constant coefficients, signature (6,-1).
FORMULA
a(n) = 3*sqrt(2)/4*((1+sqrt(2))^(2*n-1)-(1-sqrt(2))^(2*n-1)) = 6*a(n-1) - a(n-2).
G.f.: 3*x*(1-x)/(1-6*x+x^2). - Philippe Deléham, Nov 17 2008
a(n) = 3*A001653(n). - R. J. Mathar, Sep 27 2014
MATHEMATICA
CoefficientList[Series[3 (1 - x)/(1 - 6 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 11 2014 *)
LinearRecurrence[{6, -1}, {3, 15}, 20] (* Harvey P. Dale, Jun 05 2023 *)
PROG
(PARI) isok(n) = issquare(2*n^2-9); \\ Michel Marcus, Jul 10 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gregory V. Richardson, Oct 14 2002
STATUS
approved