Search: id:A077447
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%I A077447
%S A077447 4,8,16,44,92,256,536,1492,3124,8696,18208,50684,106124,295408,618536,
%T A077447 1721764,3605092,10035176,21012016,58489292,122467004,340900576,
%U A077447 713790008,1986914164,4160273044,11580584408,24247848256,67496592284
%N A077447 Numbers n such that (n^2 - 14)/2 is a square.
%C A077447 The equation "(n^2 - 14)/2 is a square" is a version of the generalized 
               Pell Equation x^2 - D*y^2 = C where x^2 - 2*y^2 = 14.
%D A077447 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 A077447 L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine 
               Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, 
               p. 341-400.
%D A077447 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 A077447 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 A077447 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PellEquation.html">; Pell Equation</a>
%F A077447 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). Lim. n -> Inf. a(n)/a(n-2) = 
               3 + 2*Sqrt(2) = RG (Grand Ratio); RG = R1*R2.
%F A077447 For n = 2*k-1, a(n) = [ 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. For n = 2*k, a(n) = [ 5*[(3+2*Sqrt(2))^n + (3-2*Sqrt(2))^n] 
               + [(3+2*Sqrt(2))^(n-1) + (3-2*Sqrt(2))^(n-1)] ] / 4. a(n) = 6*a(n-2) 
               - a(n-4)
%Y A077447 Sequence in context: A144687 A065605 A065978 this_sequence A102358 A038238 
               A023376
%Y A077447 Adjacent sequences: A077444 A077445 A077446 this_sequence A077448 A077449 
               A077450
%K A077447 nonn
%O A077447 1,1
%A A077447 Gregory V. Richardson (omomom(AT)hotmail.com), Nov 09 2002

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