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A038762
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a(n) = 6*a(n-1) - a(n-2) for n >= 2, with a(0)=3, a(1)=13.
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13
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3, 13, 75, 437, 2547, 14845, 86523, 504293, 2939235, 17131117, 99847467, 581953685, 3391874643, 19769294173, 115223890395, 671574048197, 3914220398787, 22813748344525, 132968269668363, 774995869665653
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OFFSET
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0,1
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COMMENTS
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This gives part of the (increasingly sorted) positive solutions x to the Pell equation x^2 - 2*y^2 = +7. For the y solutions see A038761. The other part of solutions is found in A101386 and A253811. - Wolfdieter Lang, Feb 05 2015
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REFERENCES
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A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.
T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, 1964, Theorem 109, pp. 207-208 with Theorem 104, pp. 197-198.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..400
M. J. DeLeon, Pell's Equation and Pell Number Triples, Fib. Quart., 14(1976), pp. 456-460.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (6,-1).
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
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Equals sqrt{2*(A038761)^2+7}.
a(n) = {13*([3+2*sqrt(2)]^n -[3-2*sqrt(2)]^n)-3*([3+2*sqrt(2)]^(n-1) - [3-2*sqrt(2)]^(n-1))}/(4*sqrt(2)).
a(n) = A077443(2n) = A038725(n)+A038725(n+1).
a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3); a(n) = (1/2)*(3+sqrt(2))*(3+2*sqrt(2))^(n-1)+(1/2)*(3-sqrt(2))*(3-2*sqrt(2))^(n-1). - Antonio Alberto Olivares, Apr 20 2008
G.f.: (3-5*x)/(1-6*x+x^2). - Philippe Deléham, Nov 03 2008, corrected by R. J. Mathar, Nov 06 2011
a(n) = -5*A001109(n) +3*A001109(n+1). - R. J. Mathar, Nov 06 2011
a(n) = rational part of z(n) = (3 + sqrt(2))*(3 + 2*sqrt(2))^n, n >= 0. z(n) gives only one part of the positive solutions to the Pell equation x^2 - 2*y^2 = 7. See the Nagell reference on how to find z(n), and a comment above. - Wolfdieter Lang, Feb 05 2015
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EXAMPLE
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a(3)^2 - 2*A038761(3)^2 = 437^2 - 2*309^2 = +7. - Wolfdieter Lang, Feb 05 2015
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MATHEMATICA
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LinearRecurrence[{6, -1}, {3, 13}, 40] (* Vincenzo Librandi, Nov 16 2011 *)
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PROG
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(MAGMA) I:=[3, 13]; [n le 2 select I[n] else 6*Self(n-1)-Self(n-2): n in [1..40]]; // Vincenzo Librandi, Nov 16 2011
(PARI) x='x+O('x^30); Vec((3-5*x)/(1-6*x+x^2)) \\ G. C. Greubel, Jul 26 2018
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CROSSREFS
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Cf. A038761, A101386, A253811.
Sequence in context: A333890 A009382 A110193 * A276894 A074517 A251658
Adjacent sequences: A038759 A038760 A038761 * A038763 A038764 A038765
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KEYWORD
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easy,nonn
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AUTHOR
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Barry E. Williams, May 03 2000
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EXTENSIONS
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More terms from James A. Sellers, May 04 2000
Unspecific Pell comment replaced by Wolfdieter Lang, Feb 05 2015
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STATUS
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approved
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