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A077447 Numbers n such that (n^2 - 14)/2 is a square. 2
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..28.

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

Lim. k -> Inf. a(2*k+1)/a(2*k) = 2.09383632135605431360 = (9 + 4*Sqrt(2))/7 = R1 (Ratio 1). Lim. k -> Inf. a(2*k)/a(2*k-1) = 2.78361162489122432754 = (11 + 6*Sqrt(2))/7 = R2 (Ratio 2). Lim. n -> Inf. 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

Sequence in context: A278377 A065605 A065978 * A337783 A301773 A102358

Adjacent sequences:  A077444 A077445 A077446 * A077448 A077449 A077450

KEYWORD

nonn

AUTHOR

Gregory V. Richardson, Nov 09 2002

STATUS

approved

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Last modified October 20 09:24 EDT 2021. Contains 348100 sequences. (Running on oeis4.)