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A097163 Expansion of (1+x-x^2)/((1-x)*(1-4*x^2)). 4

%I

%S 1,2,5,9,21,37,85,149,341,597,1365,2389,5461,9557,21845,38229,87381,

%T 152917,349525,611669,1398101,2446677,5592405,9786709,22369621,

%U 39146837,89478485,156587349,357913941,626349397,1431655765,2505397589

%N Expansion of (1+x-x^2)/((1-x)*(1-4*x^2)).

%C Interleave (4*4^n-1)/2 (see A002450) and (7*4^n-1)/3 (A206374).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-4).

%F G.f.: (1+x-x^2)/((1-x)*(1-4*x^2)).

%F a(n) = 5*2^n/4+(-2)^n/12-1/3.

%F a(n) = a(n-1)+4*a(n-2)-4*a(n-3).

%F a(2*n) = A002450(n+1).

%F a(n) = A097164(n+1)/4.

%F a(n) = (15*2^n-(-2)^n-8)/24. - _Harvey P. Dale_, Jun 17 2011

%p a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=4*a[n-2]+4 od: seq(a[n]/4, n=2..33); # _Zerinvary Lajos_, Mar 17 2008

%t CoefficientList[Series[(1+x-x^2)/((1-x)(1-4x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{1,4,-4},{1,2,5},41] (* or *) f[n_]:=(15*2^n-(-2)^n - 8)/24; Array[f, 40] (* _Harvey P. Dale_, Jun 17 2011 *)

%K nonn,easy

%O 0,2

%A _Paul Barry_, Jul 30 2004

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Last modified March 2 09:23 EST 2021. Contains 341746 sequences. (Running on oeis4.)