Search: id:A077442
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%I A077442
%S A077442 1,3,9,19,53,111,309,647,1801,3771,10497,21979,61181,128103,356589,
%T A077442 746639,2078353,4351731,12113529,25363747,70602821,147830751,411503397,
%U A077442 861620759,2398417561,5021893803,13979001969,29269742059,81475594253
%N A077442 2*n^2 + 7 is a square.
%C A077442 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).
%D A077442 L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine 
               Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, 
               p. 341-400.
%D A077442 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.
%D A077442 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.
%H A077442 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PellEquation.html">Link to a section of The World of Mathematics.</
               a>
%H A077442 J. J. O'Connor and E. F. Robertson, <a href="http://www-gap.dcs.st-and.ac.uk/
               ~history/HistTopics/Pell.html">History of Pell's Equation</a>
%H A077442 J. P. Robertson, <a href="http://members.aol.com/_ht_a/jpr2718/pell.pdf">
               Solving the Generalized Pell Equation</a>
%F A077442 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 (charliem(AT)bestweb.net), 
               Aug 14 2003
%F A077442 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 (richardchoulet(AT)yahoo.fr), Oct 11 2007
%F A077442 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 (mathar(AT)strw.leidenuniv.nl), Nov 16 2007
%Y A077442 For x an element of A077443 and y the corresponding element of this sequence, 
               the generalized Pell equation x^2 - 2*y^2 = 7 is satisfied.
%Y A077442 Cf. A077443.
%Y A077442 Sequence in context: A146901 A147477 A146677 this_sequence A147455 A146429 
               A018316
%Y A077442 Adjacent sequences: A077439 A077440 A077441 this_sequence A077443 A077444 
               A077445
%K A077442 nonn
%O A077442 0,2
%A A077442 Gregory V. Richardson (omomom(AT)hotmail.com), Nov 06 2002

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