Search: id:A077443
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%I A077443
%S A077443 3,5,13,27,75,157,437,915,2547,5333,14845,31083,86523,181165,504293,
%T A077443 1055907,2939235,6154277,17131117,35869755,99847467,209064253,
%U A077443 581953685,1218515763,3391874643,7102030325,19769294173,41393666187
%N A077443 Numbers n such that (n^2 - 7)/2 is a square.
%C A077443 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).
%C A077443 Also gives solutions >3 to the equation x^2-4 = floor(x*r*floor(x/r)) 
               where r=sqrt(2) - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 
               14 2004
%D A077443 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 A077443 L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine 
               Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, 
               p. 341-400.
%D A077443 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 A077443 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 A077443 J. P. Robertson, <a href="http://members.aol.com/_ht_a/jpr2718/pell.pdf">
               Solving the Generalized Pell Equation</a>
%H A077443 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PellEquation.html">Link to a section of The World of Mathematics.</
               a>
%F A077443 The same recurrences hold for the odd and the even indices : a(n+2)=6*a(n+1)-a(n), 
               a(n+1)=3*a(n)+2*(2*a(n)^2-14)^0.5 - Richard Choulet (richardchoulet(AT)yahoo.fr), 
               Oct 11 2007
%F A077443 O.g.f.: (-1-2*x)/(x^2-2*x-1)+(-2-x)/(x^2+2*x-1) . - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Nov 23 2007
%F A077443 a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3). - Antonio A. Olivares (olivares14031(AT)yahoo.com), 
               Apr 20 2008
%F A077443 If n is even a(n) = (1/2)*(3+sqrt(2))*(3+2*sqrt(2))^-(1/2)*n) +(1/2)*(3-sqrt(2))*(3-2*sqrt(2))^-(1/
               2)*n); if n is odd a(n) = (1/2)*(3+sqrt(2))*(3+2*sqrt(2))^((1/2)n-1/
               2)) +(1/2)*(3-sqrt(2))*(3-2*sqrt(2))^((1/2)n-1/2)) - Antonio A. Olivares 
               (olivares14031(AT)yahoo.com), Apr 20 2008
%Y A077443 If x = this sequence and y = A077442, the generalized Pell equation x^2 
               - 2*y^2 = 7 is satisfied.
%Y A077443 Cf. A038762, A077442.
%Y A077443 Sequence in context: A035082 A005198 A160823 this_sequence A147196 A110225 
               A065047
%Y A077443 Adjacent sequences: A077440 A077441 A077442 this_sequence A077444 A077445 
               A077446
%K A077443 nonn
%O A077443 1,1
%A A077443 Gregory V. Richardson (omomom(AT)hotmail.com), Nov 06 2002
%E A077443 More terms from Richard Choulet (richardchoulet(AT)yahoo.fr), Oct 11 
               2007

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