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%I #46 Jan 12 2025 14:30:48
%S 1,2,14,48,236,952,4264,18048,78736,337952,1463264,6306048,27244736,
%T 117549952,507547264,2190594048,9456660736,40819261952,176205131264,
%U 760602882048,3283257076736,14172542973952,61177656715264,264080743170048,1139938053492736,4920683538685952
%N a(n) = 2*a(n-1) + 10*a(n-2), with a(0) = 1, a(1) = 2.
%C a(n+1) = a(n) + A083101(n). A083101(n)/a(n) converges to sqrt(11).
%C Antidiagonals of A038207. - _Mark Dols_, Aug 31 2009
%C Numerators of stationary probabilities for M2/M/1 queue system. In this queue, customers arrive in groups of 2. Intensity of arrival = 2. Service rate = 5. There is only one server and an infinite queue. - _Igor Kleiner_, Nov 02 2018
%H G. C. Greubel, <a href="/A083102/b083102.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Vincenzo Librandi)
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,10).
%F G.f.: 1/(1-2*x-10*x^2).
%F From _Paul Barry_, Sep 29 2004: (Start)
%F E.g.f.: exp(x) * sinh(sqrt(11)*x) / sqrt(11).
%F a(n) = Sum_{k=0..n} binomial(n,2*k+1) * 11^k. (End)
%F a(n) = ((1+sqrt(11))^n - (1-sqrt(11))^n)/(2*sqrt(11)). - _Rolf Pleisch_, Jul 06 2009
%F G.f.: G(0)/(2-2*x), where G(k)= 1 + 1/(1 - x*(11*k-1)/( x*(11*k+10) - 1/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 14 2013
%F G.f.: Q(0)/2 , where Q(k) = 1 + 1/(1 - x*(4*k+2 + 10*x )/( x*(4*k+4 + 10*x ) + 1/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Sep 21 2013
%t CoefficientList[Series[1/(1-2x-10x^2), {x, 0, 25}], x]
%t LinearRecurrence[{2,10}, {1,2}, 30] (* _G. C. Greubel_, Jan 08 2018 *)
%o (Sage) [lucas_number1(n,2,-10) for n in range(1, 24)] # _Zerinvary Lajos_, Apr 22 2009
%o (PARI) x='x+O('x^30); Vec(1/(1-2*x-10*x^2)) \\ _G. C. Greubel_, Jan 08 2018
%o (Magma) I:=[1,2]; [n le 2 select I[n] else 2*Self(n-1) + 10*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 08 2018
%Y Cf. A038207, A083101.
%K easy,nonn
%O 0,2
%A Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003