A077447 Numbers k such that (k^2 - 14)/2 is a square.
4, 8, 16, 44, 92, 256, 536, 1492, 3124, 8696, 18208, 50684, 106124, 295408, 618536, 1721764, 3605092, 10035176, 21012016, 58489292, 122467004, 340900576, 713790008, 1986914164, 4160273044, 11580584408, 24247848256, 67496592284
Offset: 1
Keywords
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).
Crossrefs
Cf. A156649 (R1).
Programs
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Mathematica
LinearRecurrence[{0,6,0,-1},{4,8,16,44},40] (* Harvey P. Dale, Jul 22 2013 *)
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
Comments