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A106328 Numbers j such that 8*(j^2) + 9 = k^2 for some positive number k.

%I A106328 #58 Nov 26 2022 02:45:49
%S A106328 0,3,18,105,612,3567,20790,121173,706248,4116315,23991642,139833537,
%T A106328 815009580,4750223943,27686334078,161367780525,940520349072,
%U A106328 5481754313907,31950005534370,186218278892313,1085359667819508,6325939728024735,36870278700328902,214895732473948677
%N A106328 Numbers j such that 8*(j^2) + 9 = k^2 for some positive number k.
%C A106328 The ratio k(n) /(2*j(n)) tends to sqrt(2) as n increases.
%C A106328 The squares of the numbers in this sequence are one less than a triangular number: a(n)^2 = A164080(n). For example, 18^2 is 324, and 325 is a triangular number. a(n)^2 + 1 = A164055(n). a(n)^2 = A072221(n)(A072221(n)+1)/2 - 1. - _Tanya Khovanova_ & Alexey Radul, Aug 09 2009
%C A106328 For n > 0, a(n+1) is the n-th almost balancing number of first type (see Tekcan and Erdem). - _Stefano Spezia_, Nov 25 2022
%H A106328 Reinhard Zumkeller, <a href="/A106328/b106328.txt">Table of n, a(n) for n = 1..1000</a>
%H A106328 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A106328 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 A106328 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 A106328 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).
%F A106328 a(1)=0, a(2)=3 then a(n) = 6*a(n-1) - a(n-2).
%F A106328 a(n) = ((3+2*sqrt(2))^(n-1) - (3-2*sqrt(2))^(n-1))*3/4/sqrt(2). - _Max Alekseyev_, Jan 11 2007
%F A106328 a(n) = 3*A001109(n). - _M. F. Hasler_, _R. J. Mathar_, Jun 03 2009
%F A106328 a(n) = (3/4)*A005319(n-1).
%F A106328 G.f.: 3*x^2/(1 - 6*x + x^2). - _Philippe Deléham_, Nov 17 2008
%F A106328 E.g.f.: 3 - 3*exp(3*x)*(4*cosh(2*sqrt(2)*x) - 3*sqrt(2)*sinh(2*sqrt(2)*x))/4. - _Stefano Spezia_, Nov 25 2022
%t A106328 s=0;lst={};Do[s+=n;If[Sqrt[s-1]==Floor[Sqrt[s-1]],AppendTo[lst,Sqrt[s-1]]],{n,8!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Apr 02 2009 *)
%t A106328 Rest@ CoefficientList[Series[3 x^2/(1 - 6 x + x^2), {x, 0, 24}], x] (* _Michael De Vlieger_, Nov 02 2020 *)
%o A106328 (Haskell)
%o A106328 a106328 n = a106328_list !! (n-1)
%o A106328 a106328_list = 0 : 3 : zipWith (-) (map (* 6) (tail a106328_list)) a106328_list
%o A106328 -- _Reinhard Zumkeller_, Jan 10 2012
%o A106328 (PARI) concat(0, Vec(3*x^2/(1-6*x+x^2) + O(x^40))) \\ _Michel Marcus_, Sep 07 2016
%o A106328 (PARI) a(n)=([0,1;-1,6]^n*[-3;0])[1,1] \\ _Charles R Greathouse IV_, Sep 07 2016
%Y A106328 Cf. A001109, A005319, A072221, A103328, A164080, A164055.
%K A106328 nonn,easy
%O A106328 1,2
%A A106328 _Pierre CAMI_, Apr 29 2005
%E A106328 More terms from _Max Alekseyev_, Jan 11 2007