OFFSET
1,2
COMMENTS
Lim_{n->infinity} a(n)/a(n-1) = (11 + sqrt(13))/2.
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.
S. Falcon, Relationships between Some k-Fibonacci Sequences, Applied Mathematics, 2014, 5, 2226-2234; http://www.scirp.org/journal/am; http://dx.doi.org/10.4236/am.2014.515216
LINKS
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 (11,-1).
FORMULA
a(n) = ((11 + 3*sqrt(13))^n - (11 - 3*sqrt(13))^n) / ((2^n) * sqrt(13)).
From Philippe Deléham, Nov 17 2008: (Start)
a(n) = 11*a(n-1) - a(n-2) with a(1)=0 and a(2)=3.
G.f.: 3x^2/(1-11x+x^2). (End)
a(n) = A006190(2*n). - Vladimir Reshetnikov, Sep 16 2016
MATHEMATICA
LinearRecurrence[{11, -1}, {0, 3}, 20] (* Harvey P. Dale, Dec 27 2011 *)
Table[Fibonacci[2n, 3], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gregory V. Richardson, Oct 14 2002
STATUS
approved