Search: id:A077446
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%I A077446
%S A077446 1,5,11,31,65,181,379,1055,2209,6149,12875,35839,75041,208885,437371,
%T A077446 1217471,2549185,7095941,14857739,41358175,86597249,241053109,
%U A077446 504725755,1404960479,2941757281,8188709765,17145817931,47727298111
%N A077446 2*n^2 + 14 is a square.
%C A077446 The equation "2*n^2 + 14 is a square" is a version of the generalized 
               Pell Equation x^2 - D*y^2 = C where x^2 - 2*y^2 = 14.
%C A077446 Numbers n such that (ceiling(sqrt(n*n/2)))^2 = (7+n*n)/2 [From Ctibor 
               O. Zizka (c.zizka(AT)email.cz), Nov 09 2009]
%D A077446 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 A077446 L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine 
               Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, 
               p. 341-400.
%D A077446 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 A077446 J. J. O'Connor and E. F. Robertson, <a href="http://www-gap.dcs.st-and.ac.uk/
               ~history/HistTopics/Pell.html">Pell's Equation</a>
%H A077446 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PellEquation.html">; Pell Equation</a>
%F A077446 Lim. n-> Inf. a(n)/a(n-2) = 5.82842712474619009760 = 3 + 2*SQRT(2) = 
               RG (Great Ratio). Lim. k-> Inf. a(2*k+1)/a(2*k) = 2.09383632135605431360 
               = (9 + 4*SQRT(2))/7 = R1 (Ratio 1). Lim. k -> Inf. a(2*k)/a(2*k-1) 
               = 2.78361162489122432754 = (11 + 6*SQRT(2))/7 = R2 (Ratio 2); RG 
               = R1*R2. a(n) = 6*a(n-2) - a(n-4).
%F A077446 a(2*k-1) = [ 2*[(3+2*Sqrt(2))^n - (3-2*Sqrt(2))^n] - [(3+2*Sqrt(2))^(n-1) 
               - (3-2*Sqrt(2))^(n-1)] + [(3+2*Sqrt(2))^(n-2) - (3-2*Sqrt(2))^(n-2)] 
               ] / (4*Sqrt(2)) a(2*k) = [ 5*[(3+2*Sqrt(2))^n - (3-2*Sqrt(2))^n] 
               + [(3+2*Sqrt(2))^(n-1) - (3-2*Sqrt(2))^(n-1)] ] / (4*Sqrt(2)). a(n) 
               = 6*a(n-2) - a(n-4)
%Y A077446 2*(a(n))^2 + 14 = (A077447)^2.
%Y A077446 Sequence in context: A057470 A038580 A106088 this_sequence A023276 A074648 
               A106908
%Y A077446 Adjacent sequences: A077443 A077444 A077445 this_sequence A077447 A077448 
               A077449
%K A077446 nonn
%O A077446 1,2
%A A077446 Gregory V. Richardson (omomom(AT)hotmail.com), Nov 09 2002

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