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A004298 Expansion of (1+2*x+x^2)/(1-66*x+x^2).

%I A004298 #23 Dec 28 2020 20:31:55
%S A004298 1,68,4488,296140,19540752,1289393492,85080429720,5614018968028,
%T A004298 370440171460128,24443437297400420,1612896421456967592,
%U A004298 106426720378862460652,7022550648583465435440,463381916086129856278388,30576183911035987048938168,2017564756212289015373640700
%N A004298 Expansion of (1+2*x+x^2)/(1-66*x+x^2).
%D A004298 P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 160, middle display.
%H A004298 Vincenzo Librandi, <a href="/A004298/b004298.txt">Table of n, a(n) for n = 0..500</a>
%H A004298 J. M. Alonso, <a href="http://dx.doi.org/10.1007/978-1-4612-3142-4_1">Growth functions of amalgams</a>, in Alperin, ed., Arboreal Group Theory, Springer, pp. 1-34, esp. p. 32.
%H A004298 Hacène Belbachir, Soumeya Merwa Tebtoub, 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 A004298 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (66,-1).
%F A004298 For n > 0, a(n) = 68*A097316(n-1). - _Gerald McGarvey_, Jun 16 2007
%F A004298 From _Colin Barker_, Apr 16 2016: (Start)
%F A004298 a(n) = (sqrt(17)*(33+8*sqrt(17))^(-n)*(-1+(33+8*sqrt(17))^(2*n)))/4 for n>0.
%F A004298 a(n) = 66*a(n-1) - a(n-2) for n>2.
%F A004298 (End)
%F A004298 a(n) = (-(-1)^(2^n) + sqrt(17)*sinh(n*log(33+8*sqrt(17))) + 1)/2. - _Ilya Gutkovskiy_, Apr 16 2016
%t A004298 CoefficientList[Series[(1+2*x+x^2)/(1-66*x+x^2),{x,0,50}],x] (* _Vincenzo Librandi_, Feb 25 2012 *)
%t A004298 LinearRecurrence[{66,-1},{1,68,4488},20] (* _Harvey P. Dale_, Sep 23 2020 *)
%o A004298 (PARI) Vec((1+2*x+x^2)/(1-66*x+x^2) + O(x^50)) \\ _Colin Barker_, Apr 16 2016
%Y A004298 Pairwise sums of A078989.
%Y A004298 Cf. A097316.
%K A004298 nonn,easy
%O A004298 0,2
%A A004298 _N. J. A. Sloane_