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 A038762 a(n) = 6*a(n-1) - a(n-2) for n >= 2, with a(0)=3, a(1)=13. 13
 3, 13, 75, 437, 2547, 14845, 86523, 504293, 2939235, 17131117, 99847467, 581953685, 3391874643, 19769294173, 115223890395, 671574048197, 3914220398787, 22813748344525, 132968269668363, 774995869665653 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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 REFERENCES 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. LINKS 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). FORMULA 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 EXAMPLE a(3)^2 - 2*A038761(3)^2 = 437^2 - 2*309^2 = +7. - Wolfdieter Lang, Feb 05 2015 MATHEMATICA LinearRecurrence[{6, -1}, {3, 13}, 40] (* Vincenzo Librandi, Nov 16 2011 *) PROG (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 CROSSREFS Cf. A038761, A101386, A253811. Sequence in context: A189886 A009382 A110193 * A276894 A074517 A251658 Adjacent sequences:  A038759 A038760 A038761 * A038763 A038764 A038765 KEYWORD easy,nonn AUTHOR Barry E. Williams, May 03 2000 EXTENSIONS More terms from James A. Sellers, May 04 2000 Unspecific Pell comment replaced by Wolfdieter Lang, Feb 05 2015 STATUS approved

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Last modified December 12 00:07 EST 2018. Contains 318052 sequences. (Running on oeis4.)