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A075841
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Numbers k such that 2*k^2 - 9 is a square.
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7
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3, 15, 87, 507, 2955, 17223, 100383, 585075, 3410067, 19875327, 115841895, 675176043, 3935214363, 22936110135, 133681446447, 779152568547, 4541233964835, 26468251220463, 154268273357943, 899141388927195
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OFFSET
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1,1
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COMMENTS
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Lim. n-> Inf. a(n)/a(n-1) = 3 + 2*sqrt(2).
Positive values of x (or y) satisfying x^2 - 6*x*y + y^2 + 36 = 0. - Colin Barker, Feb 08 2014
For each member t of the sequence there exists a nonnegative r such that t^2 = r^2 + (r+3)^2. The r values are in A241976. Example: 87^2 = 60^2 + 63^2. - Bruno Berselli, Jul 10 2017
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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, 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.
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LINKS
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FORMULA
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a(n) = 3*sqrt(2)/4*((1+sqrt(2))^(2*n-1)-(1-sqrt(2))^(2*n-1)) = 6*a(n-1) - a(n-2).
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MATHEMATICA
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CoefficientList[Series[3 (1 - x)/(1 - 6 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 11 2014 *)
LinearRecurrence[{6, -1}, {3, 15}, 20] (* Harvey P. Dale, Jun 05 2023 *)
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PROG
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(PARI) isok(n) = issquare(2*n^2-9); \\ Michel Marcus, Jul 10 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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