%I #38 Nov 26 2022 02:47:22
%S 3,9,51,297,1731,10089,58803,342729,1997571,11642697,67858611,
%T 395508969,2305195203,13435662249,78308778291,456417007497,
%U 2660193266691,15504742592649,90368262289203,526704831142569,3069860724566211,17892459516254697,104284896372961971
%N Numbers k such that k^2 = 8*j^2 + 9.
%C The ratio a(n)/(2*j(n)) tends to sqrt(2) as n increases.
%C After 3, first differences of A301383. - _Bruno Berselli_, Mar 22 2018
%C For n > 0, a(n+1) is the n-th almost Lucas-balancing number of first type (see Tekcan and Erdem). - _Stefano Spezia_, Nov 25 2022
%H Colin Barker, <a href="/A106329/b106329.txt">Table of n, a(n) for n = 1..1000</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Soumeya M. Tebtoub, Hacène Belbachir, and László Németh, <a href="https://hal.archives-ouvertes.fr/hal-02918958/document#page=18">Integer sequences and ellipse chains inside a hyperbola</a>, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 17-18.
%H Ahmet Tekcan and Alper Erdem, <a href="https://arxiv.org/abs/2211.08907">General Terms of All Almost Balancing Numbers of First and Second Type</a>, arXiv:2211.08907 [math.NT], 2022.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).
%F a(1)=3, a(2)=9 then a(n) = 6*a(n-1)-a(n-2).
%F G.f.: 3*x*(1 - 3*x)/(1 - 6*x + x^2). - _Philippe Deléham_, Nov 17 2008
%F a(n) = (3/2)*A003499(n-1).
%F a(n) = 3*((3-2*sqrt(2))^(n-1) + (3+2*sqrt(2))^(n-1))/2. - _Colin Barker_, Oct 13 2015
%F E.g.f.: 3*exp(3*x)*(3*cosh(2*sqrt(2)*x) - 2*sqrt(2)*sinh(2*sqrt(2)*x)) - 9. - _Stefano Spezia_, Nov 25 2022
%t CoefficientList[Series[3 x (1 - 3 x)/(1 - 6 x + x^2), {x, 0, 23}], x] (* _Michael De Vlieger_, Nov 02 2020 *)
%o (PARI) Vec((3-9*x)/(1-6*x+x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Dec 28 2011
%Y Cf. A003499, A103328, A301383.
%K nonn,easy
%O 1,1
%A _Pierre CAMI_, Apr 29 2005