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%I #56 Mar 23 2024 07:24:54
%S 4,20,116,676,3940,22964,133844,780100,4546756,26500436,154455860,
%T 900234724,5246952484,30581480180,178241928596,1038870091396,
%U 6054978619780,35291001627284,205691031143924,1198855185236260
%N Numbers k such that (k^2 - 8)/2 is a square.
%C The equation "(k^2 - 8)/2 is a square" is a version of the generalized Pell Equation "x^2 - D*y^2 = C".
%C a(n)^2 - 2*A077444(n) = 8.
%C From _Wolfdieter Lang_, Jan 18 2013: (Start)
%C 4*(1-z)/(1-6*z+z^2) = Sum_{n>=0} a(n+1)*z^n is the formal power series for tan(4*x)/tan(x) if one lets
%C z = (tan(x))^2. For the numerator and denominator of this o.g.f. see A034867 and A034839, respectively. Convergence holds for 0 <= z < 3 - 2*sqrt(2), approximately 0.1715728753. This means for |x| < Pi/8, approximately 0.3926990818.
%C See also the o.g.f. given by _Johannes W. Meijer_, Aug 01 2010, in the formula section of A001653 = (this sequence)/4.
%C (End)
%C Positive values of x (or y) satisfying x^2 - 6*x*y + y^2 + 64 = 0. - _Colin Barker_, Feb 13 2014
%D 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 L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400.
%D Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, pp. 139-147.
%H Vincenzo Librandi, <a href="/A077445/b077445.txt">Table of n, a(n) for n = 1..1000</a>
%H 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 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Klaus Nagel, <a href="/A077445/a077445.pdf">Length of cycles in bijection of 2 quadratic grids rotated by Pi/6 relative to each other</a>, (2009).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PellEquation.html">Pell Equation</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).
%F 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.
%F a(n) = 6*a(n-1) - a(n-2) = 4*A001653(n).
%F G.f.: 4*(x-x^2)/(1-6*x+x^2).
%F With a=3+2*sqrt(2), b=3-2*sqrt(2): a(n) = sqrt(2)*(a^((2*n-1)/2) + b^((2*n-1)/2)). a(n) = sqrt(2*A003499(2*n-1)+4). - Mario Catalani (mario.catalani(AT)unito.it), Mar 24 2003
%F a(n) = (A003499(n+1) + A003499(n))/2. - Mario Catalani (mario.catalani(AT)unito.it), Mar 31 2003
%F a(n) = (2 + sqrt(2))*(3 + 2*sqrt(2))^n + (2 - sqrt(2))*(3- 2*sqrt(2))^n. - _Antonio Alberto Olivares_, Feb 23 2006
%F a(n) = 2*A075870(n). - _Bruno Berselli_, Nov 27 2013
%F G.f.: 2*Q(0)*x*(1-x)/(1-3*x), where Q(k) = 1 + 1/( 1 - x*(8*k-9)/( x*(8*k-1) - 3/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Dec 10 2013
%t CoefficientList[Series[4 (1 - x)/(1 - 6 x + x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 14 2014 *)
%o (PARI) a(n)=if(n<1,0,subst(poltchebi(n)+poltchebi(n-1),x,3))
%Y Cf. A001653, A003499, A075870.
%K nonn,easy
%O 1,1
%A _Gregory V. Richardson_, Nov 09 2002
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