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A137255 a(n) = 2*a(n-1) + 4*a(n-2) - 6*a(n-3) - 3*a(n-4) for n > 3, with a(0)=1, a(1)=2, a(2)=4, a(3)=8.

%I A137255 #15 Oct 03 2021 09:20:29
%S A137255 1,2,4,8,17,36,80,178,409,942,2212,5204,12377,29472,70592,169198,
%T A137255 406801,978426,2357092,5679488,13696385,33032892,79703120,192321034,
%U A137255 464168041,1120302822,2704242244,6527724428,15758096777,38040729336,91834772480
%N A137255 a(n) = 2*a(n-1) + 4*a(n-2) - 6*a(n-3) - 3*a(n-4) for n > 3, with a(0)=1, a(1)=2, a(2)=4, a(3)=8.
%H A137255 G. C. Greubel, <a href="/A137255/b137255.txt">Table of n, a(n) for n = 0..1000</a>
%H A137255 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,4,-6,-3).
%F A137255 a(n) = (3/8)*3^(n/2)*(1 + (-1)^n) + (5/24)*3^((n+1)/2)*(1 - (-1)^n) + (1/8)*(1+sqrt(2))^(n+1) + (1/8)*(1-sqrt(2))^(n+1). - _Emeric Deutsch_, Mar 31 2008
%F A137255 G.f.: (1 - 4*x^2 - 2*x^3)/(( 1 -3*x^2) *(1 -2*x -x^2)). - _Harvey P. Dale_, May 03 2018
%F A137255 4*a(n) = A078057(n) + A083658(n+2). - _R. J. Mathar_, Oct 03 2021
%p A137255 a:=proc(n) options operator, arrow: expand((3/8)*3^((1/2)*n)*(1+(-1)^n)+(5/24)*3^((1/2)*n+1/2)*(1-(-1)^n)+(1/8)*(1+sqrt(2))^(n+1)+(1/8)*(1-sqrt(2))^(n+1)) end proc: seq(a(n),n=0..30); # _Emeric Deutsch_, Mar 31 2008
%t A137255 LinearRecurrence[{2,4,-6,-3},{1,2,4,8},40] (* or *) CoefficientList[ Series[ (1-4 x^2-2 x^3)/(1-2 x-4 x^2+6 x^3+3 x^4),{x,0,40}],x] (* _Harvey P. Dale_, May 03 2018 *)
%o A137255 (Magma)
%o A137255 R<x>:=PowerSeriesRing(Integers(), 40);
%o A137255 Coefficients(R!( (1-4*x^2-2*x^3)/(1-2*x-4*x^2+6*x^3+3*x^4) )); // _G. C. Greubel_, Apr 11 2021
%o A137255 (Sage)
%o A137255 def A137255_list(prec):
%o A137255     P.<x> = PowerSeriesRing(ZZ, prec)
%o A137255     return P( (1-4*x^2-2*x^3)/(1-2*x-4*x^2+6*x^3+3*x^4) ).list()
%o A137255 A137255_list(40) # _G. C. Greubel_, Apr 11 2021
%K A137255 nonn
%O A137255 0,2
%A A137255 _Paul Curtz_, Mar 11 2008
%E A137255 More terms from _Emeric Deutsch_, Mar 31 2008