OFFSET
1,2
COMMENTS
(5/4)*a(n)^2 +1 is a triangular number. - Bruno Berselli, Aug 17 2013
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..1000
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 (0,0,38,0,0,-1).
FORMULA
From Gregory V. Richardson, Oct 16 2002: (Start)
Limit_{n->oo} a(n)/a(n-3) = 19 + 6*sqrt(10).
Limit_{n->oo} a(3*n)/a(3*n-1) = (11 + 2*sqrt(10))/9.
Limit_{n->oo} a(3*n+1)/a(3*n) = (7 + 2*sqrt(10))/3.
Limit_{n->oo} a(3*n+2)/a(3*n+1) = (7 + 2*sqrt(10))/3. (End)
G.f.: 2*x^2*(1+2*x+9*x^2+2*x^3+x^4) / ( 1-38*x^3+x^6 ). - R. J. Mathar, Jul 03 2011
a(n) = 2*A075873(n). - R. J. Mathar, Jul 03 2011
MATHEMATICA
CoefficientList[Series[2 x (1 + 2 x + 9 x^2 + 2 x^3 + x^4) / (1 - 38 x^3 + x^6), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 17 2013 *)
PROG
(Magma) I:=[0, 2, 4, 18, 80, 154]; [n le 6 select I[n] else 38*Self(n-3)-Self(n-6): n in [1..30]]; // Vincenzo Librandi, Aug 17 2013
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Gregory V. Richardson, Oct 14 2002
STATUS
approved