Search: id:A077444
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%I A077444
%S A077444 2,14,82,478,2786,16238,94642,551614,3215042,18738638,109216786,
%T A077444 636562078,3710155682,21624372014,126036076402,734592086398,
%U A077444 4281516441986,24954506565518,145445522951122,847718631141214
%N A077444 Numbers n such that (n^2 + 4)/2 is a square.
%C A077444 The equation "(n^2 + 4)/2 is a square" is a version of the generalized 
               Pell Equation x^2 - D*y^2 = C where x^2 - 2*y^2 = -4.
%C A077444 Sequence of all positive integers k such that continued fraction [k,k,
               k,k,k,k,...] belongs to Q(sqrt(2)). - Thomas Baruchel Sep 15 2003
%C A077444 Equivalently, 2*n^2 + 8 is a square.
%C A077444 Numbers n such that (ceiling(sqrt(n*n/2)))^2 = 2 + n*n/2 [From Ctibor 
               O. Zizka (c.zizka(AT)email.cz), Nov 09 2009]
%D A077444 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 A077444 L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine 
               Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, 
               p. 341-400.
%D A077444 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 A077444 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A077444 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A077444 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 A077444 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PellEquation.html">; Pell Equation</a>
%H A077444 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               NSWNumber.html">NSW Number</a>
%F A077444 a(n) =[ [(3+2*Sqrt(2))^n - (3-2*Sqrt(2))^n] + [(3+2*Sqrt(2))^(n-1) - 
               3-2*Sqrt(2))^(n-1)] ] / (2*Sqrt(2))
%F A077444 Recurrence: a(n) = 6*a(n-1) - a(n-2), starting 2, 14.
%F A077444 Offset 0, with a=3+2sqrt(2), b=3-2sqrt(2): a(n)=a^((2n+1)/2)-b^((2n+1)/
               2). a(n)=2(A001109(n+1)+A001109(n))=(A003499(n+1)-A003499(n))/2=2sqrt(A001108(2n+1)) 
               =sqrt(A003499(2n+1)-2). - Mario Catalani (mario.catalani(AT)unito.it), 
               Mar 31 2003
%F A077444 Lim. n -> Inf. a(n)/a(n-1) = 5.82842712474619009760 = 3 + 2*Sqrt(2).
%F A077444 a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3), a(0) = 2, a(1) = 14; a(3) = 82; 
               a(n) = (1+SQRT(2))^(2N+1) + (1-SQRT(2))^(2N+1)
%F A077444 G.f.: 2*x*(1+x)/(1-6*x+x^2). a(n) = 2*[7*A001109(n)-A001109(n+1)]. - 
               R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 16 2007
%Y A077444 Equals 2 * A002315.
%Y A077444 (A077445(n))^2 - 2*a(n) = 8.
%Y A077444 First differences of A001541. Pairwise sums of A001542. Bisection of 
               A002203 and A080039.
%Y A077444 Sequence in context: A026291 A102401 A077461 this_sequence A138126 A053141 
               A036692
%Y A077444 Adjacent sequences: A077441 A077442 A077443 this_sequence A077445 A077446 
               A077447
%K A077444 nonn
%O A077444 1,1
%A A077444 Gregory V. Richardson (omomom(AT)hotmail.com), Nov 09 2002

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