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%I #35 Sep 08 2022 08:45:09
%S 0,8,48,280,1632,9512,55440,323128,1883328,10976840,63977712,
%T 372889432,2173358880,12667263848,73830224208,430314081400,
%U 2508054264192,14618011503752,85200014758320,496582077046168,2894292447518688
%N a(n) = sqrt(2)*( (3+2*sqrt(2))^n - (3-2*sqrt(2))^n ).
%C Numbers m such that ceiling( sqrt(m*m/2) )^2 = 4 + m*m/2. - _Ctibor O. Zizka_, Nov 09 2009
%C Numbers m such that 2*m^2+16 is a square. - _Bruno Berselli_, Dec 17 2014
%H G. C. Greubel, <a href="/A081554/b081554.txt">Table of n, a(n) for n = 0..1000</a>
%H Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Nemeth/nemeth7.html">Ellipse Chains and Associated Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
%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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).
%F a(n)^2 = 2*A003499(n)^2 - 8.
%F a(n) = 8*A001109(n).
%F G.f.: 8*x/(1-6*x+x^2). - _Philippe Deléham_, Nov 17 2008
%F a(0)=0, a(1)=8, a(n) = 6*a(n-1) - a(n-2) for n>1. - _Philippe Deléham_, Sep 19 2009
%F a(n) = 4*Pell(2*n) = 4*A000129(2*n). - _G. C. Greubel_, Aug 16 2018
%t a = 3 + 2Sqrt[2]; b = 3 - 2Sqrt[2]; Table[Simplify[Sqrt[2](a^n - b^n)], {n, 0, 25}]
%t CoefficientList[Series[8x/(1-6x+x^2),{x,0,40}],x] (* _Harvey P. Dale_, Mar 11 2011 *)
%t Table[4 Fibonacci[2 n, 2], {n, 0, 50}] (* _G. C. Greubel_, Aug 16 2018 *)
%o (PARI) x='x+O('x^50); concat([0], Vec(8*x/(1-6*x+x^2))) \\ _G. C. Greubel_, Aug 16 2018
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(8*x/(1-6*x+x^2))); // _G. C. Greubel_, Aug 16 2018
%K nonn,easy
%O 0,2
%A Mario Catalani (mario.catalani(AT)unito.it), Mar 21 2003
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