OFFSET
1,1
COMMENTS
The equation "(n^2 - 14)/2 is a square" is a version of the generalized Pell Equation x^2 - D*y^2 = C where x^2 - 2*y^2 = 14.
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
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 (0,6,0,-1).
FORMULA
Limit_{k -> oo} a(2*k+1)/a(2*k) = 2.09383632135605431360 = (9 + 4*sqrt(2))/7 = R1.
Limit_{k -> oo} a(2*k)/a(2*k-1) = 2.78361162489122432754 = (11 + 6*sqrt(2))/7 = R2.
Limit_{k -> oo} a(n)/a(n-2) = 3 + 2*sqrt(2) = RG (Grand Ratio); RG = R1*R2.
For n = 2*k-1, a(n) = [ 2*[(3+2*sqrt(2))^n + (3-2*sqrt(2))^n] - [(3+2*sqrt(2))^(n-1) + (3-2*sqrt(2))^(n-1)] + [(3+2*sqrt(2))^(n-2) + (3-2*sqrt(2))^(n-2)] ] / 4.
For n = 2*k, a(n) = [ 5*[(3+2*sqrt(2))^n + (3-2*sqrt(2))^n] + [(3+2*sqrt(2))^(n-1) + (3-2*sqrt(2))^(n-1)] ] / 4.
a(n) = 6*a(n-2) - a(n-4) = 4*A006452(n).
G.f.: -4*x*(x-1)*(x^2+3*x+1) / ( (x^2+2*x-1)*(x^2-2*x-1) ). - R. J. Mathar, Jul 03 2011
MATHEMATICA
LinearRecurrence[{0, 6, 0, -1}, {4, 8, 16, 44}, 40] (* Harvey P. Dale, Jul 22 2013 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gregory V. Richardson, Nov 09 2002
STATUS
approved