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A080882 a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0)=1, a(1)=3, a(2)=7.

%I A080882 #23 Mar 23 2025 18:12:30
%S A080882 1,3,7,22,52,164,388,1224,2896,9136,21616,68192,161344,508992,1204288,
%T A080882 3799168,8988928,28357376,67094272,211662336,500798464,1579869184,
%U A080882 3738010624,11792304128,27900891136,88018956288,208255086592
%N A080882 a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0)=1, a(1)=3, a(2)=7.
%H A080882 Harvey P. Dale, <a href="/A080882/b080882.txt">Table of n, a(n) for n = 0..1000</a>
%H A080882 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0, 8, 0, -4).
%F A080882 a(2n)=A080879(2n+1)=A080876(2n+4)/4, a(2n+1)=A080879(2n+2)/2=A080876(2n+5)/4.
%F A080882 G.f.: (-2*x^3 - x^2 + 3*x + 1)/(4*x^4 - 8*x^2 + 1).
%F A080882 a(n + 4) = 8*a(n + 2) - 4*a(n). - _Richard Choulet_, Dec 06 2008
%F A080882 a(n) = (7/24*3^(1/2) + 1/2)*((1 + sqrt(3)))^n + ( - 7/24*3^(1/2) + 1/2)*((1 - sqrt(3)))^n + ( - 1/24*3^(1/2))*( - (1 + sqrt(3)))^n + (1/24*3^(1/2))*( - ((1 - sqrt(3))))^n. - _Richard Choulet_, Dec 06 2008
%p A080882 a:= n-> (Matrix([[22,7,3,1]]). Matrix(4, (i,j)-> if (i=j-1) then 1 elif j=1 then [0,8,0,-4][i] else 0 fi)^(n))[1,4]: seq(a(n), n=0..26); # _Alois P. Heinz_, Aug 23 2008
%t A080882 a[0]=1; a[1]=3; a[2]=7; a[3]=22; a[n_] := a[n] = 8*a[n-2] - 4*a[n-4]; Table[a[n], {n, 0, 26}] (* _Jean-François Alcover_, Jun 15 2015, after _Richard Choulet_ *)
%t A080882 LinearRecurrence[{0,8,0,-4},{1,3,7,22},30] (* _Harvey P. Dale_, Mar 23 2025 *)
%Y A080882 Cf. A080876, A080877, A080878, A080879, A080880, A080881.
%K A080882 nonn
%O A080882 0,2
%A A080882 _Paul D. Hanna_, Feb 22 2003