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%I #38 Oct 24 2023 23:15:14
%S 1,20,360,6460,115920,2080100,37325880,669785740,12018817440,
%T 215668928180,3870021889800,69444725088220,1246135029698160,
%U 22360985809478660,401251609540917720,7200167985927040300
%N Expansion of (1+x)^2/(1-18*x+x^2).
%D J. M. Alonso, Growth functions of amalgams, in Alperin, ed., Arboreal Group Theory, Springer, pp. 1-34, esp. p. 32.
%D P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 160, middle display.
%H Vincenzo Librandi, <a href="/A004292/b004292.txt">Table of n, a(n) for n = 0..800</a>
%H 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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (18,-1).
%F a(n) = (1/2)*(1 - (-1)^2^n + (20+9*sqrt(5))*((9+4*sqrt(5))^(2*n) - 1)/(9+4*sqrt(5))^(n+1)). - _Gerry Martens_, May 30 2015
%p f:= gfun:-rectoproc({a(n)=18*a(n-1)-a(n-2),a(0)=1,a(1)=20,a(2)=360},a(n),remember):
%p map(f, [$0..20]); # _Robert Israel_, Jun 01 2015
%t CoefficientList[Series[(1+x)^2/(1-18*x+x^2),{x,0,20}],x] (* _Vincenzo Librandi_, Jun 13 2012 *)
%t a[n_]:=1/2(1-(-1)^2^n+(20+9 Sqrt[5])((9+4 Sqrt[5])^(2 n)-1)/(9+4 Sqrt[5])^(n+1));Table[a[n] // FullSimplify,{n,0,20}] (* _Gerry Martens_, May 30 2015 *)
%o (PARI) Vec((1+x)^2/(1-18*x+x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%Y Pairwise sums of A049629.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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