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Search: id:A077447
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| A077447 |
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Numbers n such that (n^2 - 14)/2 is a square. |
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+0
2
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| 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
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OFFSET
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1,1
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COMMENT
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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.
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REFERENCES
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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, p. 341-400.
Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, p. 139-147.
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LINKS
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J. J. O'Connor and E. F. Robertson, Pell's Equation
Eric Weisstein's World of Mathematics, ; Pell Equation
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FORMULA
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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)
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CROSSREFS
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Sequence in context: A144687 A065605 A065978 this_sequence A102358 A038238 A023376
Adjacent sequences: A077444 A077445 A077446 this_sequence A077448 A077449 A077450
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KEYWORD
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nonn
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AUTHOR
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Gregory V. Richardson (omomom(AT)hotmail.com), Nov 09 2002
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