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Expansion of (1-x+2*x^2)/((1-x)*(1-x-2*x^2)).
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%I #30 Jan 25 2023 11:51:56

%S 1,1,5,9,21,41,85,169,341,681,1365,2729,5461,10921,21845,43689,87381,

%T 174761,349525,699049,1398101,2796201,5592405,11184809,22369621,

%U 44739241,89478485,178956969,357913941,715827881,1431655765,2863311529

%N Expansion of (1-x+2*x^2)/((1-x)*(1-x-2*x^2)).

%C Partial sums of A097073.

%C This is the sequence A(1,1;1,2;2) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. [_Wolfdieter Lang_, Oct 18 2010]

%H G. C. Greubel, <a href="/A097074/b097074.txt">Table of n, a(n) for n = 0..1000</a>

%H Wolfdieter Lang, <a href="/A097074/a097074.pdf">Notes on certain inhomogeneous three term recurrences.</a> [_Wolfdieter Lang_, Oct 18 2010]

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2).

%F a(n) = 2*A001045(n+1) - 1.

%F a(n) = (2^(n+2) + 2*(-1)^n - 3)/3.

%F From _Wolfdieter Lang_, Oct 18 2010: (Start)

%F a(n) = a(n-1) + 2*a(n-2) + 2, a(0)=1, a(1)=1.

%F a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), a(0)=1=a(1), a(2)=5. Observed by G. Detlefs. See the W. Lang link. (End)

%F a(n) = 3*a(n-1) - 2*a(n-2) + 4*(-1)^n. - _Gary Detlefs_, Dec 19 2010

%F a(n) = A000975(n+1) - A000975(n) + 2*A000975(n-1). - _R. J. Mathar_, Feb 27 2019

%F E.g.f.: (1/3)*(2*exp(-x) - 3*exp(x) + 4*exp(2*x)). - _G. C. Greubel_, Aug 18 2022

%t CoefficientList[Series[(1-x+2x^2)/((1-x)(1-x-2x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{2,1,-2},{1,1,5},40] (* _Harvey P. Dale_, Apr 09 2018 *)

%o (Magma) [(2^(n+2) +2*(-1)^n -3)/3: n in [0..40]]; // _G. C. Greubel_, Aug 18 2022

%o (SageMath) [(2^(n+2) +2*(-1)^n -3)/3 for n in (0..40)] # _G. C. Greubel_, Aug 18 2022

%Y Cf. A000975, A001045, A097073.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Jul 22 2004

%E Correction of the homogeneous recurrence and index link added by _Wolfdieter Lang_, Nov 16 2013