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 A077442 2*a(n)^2 + 7 is a square. 12
 1, 3, 9, 19, 53, 111, 309, 647, 1801, 3771, 10497, 21979, 61181, 128103, 356589, 746639, 2078353, 4351731, 12113529, 25363747, 70602821, 147830751, 411503397, 861620759, 2398417561, 5021893803, 13979001969, 29269742059, 81475594253 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Lim. n -> Inf. a(n)/a(n-2) = 3 + 2*Sqrt(2) = R1*R2. Lim. k -> Inf. a(2*k-1)/a(2*k) = (9 + 4*Sqrt(2))/7 = R1 (ratio #1). Lim. k -> Inf. a(2*k)/a(2*k-1) = (11 + 6*Sqrt(2))/7 = R2 (ratio #2). a(n) gives for n >= 0 all positive y-values solving the (generalized) Pell equation x^2 - 2*y^2 = 7. A077443(n+1) gives the corresponding x-values. See, e.g., the Nagell reference on how to find all solutions. - Wolfdieter Lang, Feb 05 2015 REFERENCES L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400. 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. 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. T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, 1964, Theorem 109, pp. 207-208 with Theorem 104, pp. 197-198. LINKS Colin Barker, Table of n, a(n) for n = 0..1000 J. J. O'Connor and E. F. Robertson, History of Pell's Equation J. P. Robertson, Solving the Generalized Pell Equation Eric Weisstein's World of Mathematics, Pell Equation. Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1). FORMULA For n>0, a(2n) = A046090(n) + A001653(n) + A001652(n-1); a(2n+1) = A001652(n+1) - A001652(n-1) - A001653(n-1); e.g. 53=21+29+3; 111=119-3-5. - Charlie Marion, Aug 14 2003 The same recurrences hold for the odd and even indices respectively : a(n+2) = 6*a(n+1) - a(n), a(n+1) = 3*a(n) + 2*(2*a(n)^2+7)^0.5. - Richard Choulet, Oct 11 2007 G.f.: (x+1)^3/(x^2+2*x-1)/(x^2-2*x-1). a(n)= [ -A077985(n)-3*A077985(n-1)+3*A000129(n+1)+A000129(n)]/2. - R. J. Mathar, Nov 16 2007 a(n) = 6*a(n-2) - a(n-4) with a(1)=1, a(2)=3, a(3)=9, a(4)=19. - Sture Sjöstedt, Oct 08 2012 a(n) = ((-(-1 - sqrt(2))^n*(-2+sqrt(2)) - (-1+sqrt(2))^n*(2+sqrt(2)) + (1-sqrt(2))^n*(-4+3*sqrt(2)) + (1+sqrt(2))^n*(4+3*sqrt(2))))/(4*sqrt(2)). - Colin Barker, Mar 27 2016 EXAMPLE a(4)^2 - 2*a(3)^2 = 27^2 - 2*19^2  = +7. - Wolfdieter Lang, Feb 05 2015 MATHEMATICA CoefficientList[Series[(1+3 x+3 x^2+x^3)/ (1-6 x^2+x^4), {x, 0, 50}], x]  (* Harvey P. Dale, Mar 12 2011 *) LinearRecurrence[{0, 6, 0, -1}, {1, 3, 9, 19}, 50] (* Sture Sjöstedt, Oct 08 2012 *) PROG (PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -1, 0, 6, 0]^n*[1; 3; 9; 19])[1, 1] \\ Charles R Greathouse IV, Jun 20 2015 (PARI) Vec((x+1)^3/(x^2+2*x-1)/(x^2-2*x-1) + O(x^50)) \\ Colin Barker, Mar 27 2016 CROSSREFS Cf. A077443, A038762, A038761, A101386, A253811. Sequence in context: A146901 A147477 A146677 * A147455 A146429 A297390 Adjacent sequences:  A077439 A077440 A077441 * A077443 A077444 A077445 KEYWORD nonn,easy AUTHOR Gregory V. Richardson, Nov 06 2002 EXTENSIONS Edited: n in Name replaced by a(n). Pell comments moved to comment section. - Wolfdieter Lang, Feb 05 2015 STATUS approved

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Last modified July 25 02:40 EDT 2021. Contains 346276 sequences. (Running on oeis4.)