cp's OEIS Frontend

A004294 Expansion of (1+2*x+x^2)/(1-34*x+x^2).

%I A004294 #22 Dec 28 2020 20:31:46
%S A004294 1,36,1224,41580,1412496,47983284,1630019160,55372668156,
%T A004294 1881040698144,63900011068740,2170719335639016,73740557400657804,
%U A004294 2505008232286726320,85096539340348037076,2890777329339546534264,98201332658204234127900,3335954533049604413814336
%N A004294 Expansion of (1+2*x+x^2)/(1-34*x+x^2).
%D A004294 P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 160, middle display.
%H A004294 Vincenzo Librandi, <a href="/A004294/b004294.txt">Table of n, a(n) for n = 0..600</a>
%H A004294 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 A004294 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 A004294 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (34,-1).
%F A004294 From _Colin Barker_, Apr 16 2016: (Start)
%F A004294 a(n) = 3*((17+12*sqrt(2))^(1-n)*(-1+(17+12*sqrt(2))^(2*n)))/(48+34*sqrt(2)) for n>0.
%F A004294 a(n) = 34*a(n-1) - a(n-2) for n>2.
%F A004294 (End)
%F A004294 a(n) = (-(-1)^(2^n) + 3*sqrt(2)*sinh(n*log(17+12*sqrt(2))) + 1)/2. - _Ilya Gutkovskiy_, Apr 16 2016
%t A004294 CoefficientList[Series[(1+2*x+x^2)/(1-34*x+x^2),{x,0,20}],x] (* _Vincenzo Librandi_, Jun 14 2012 *)
%t A004294 LinearRecurrence[{34,-1},{1,36,1224},20] (* _Harvey P. Dale_, Mar 29 2019 *)
%o A004294 (PARI) Vec((1+2*x+x^2)/(1-34*x+x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%Y A004294 Pairwise sums of A046176.
%K A004294 nonn,easy
%O A004294 0,2
%A A004294 _N. J. A. Sloane_