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%I #18 Jul 11 2023 19:35:02
%S 1,4,8,20,36,84,148,340,596,1364,2388,5460,9556,21844,38228,87380,
%T 152916,349524,611668,1398100,2446676,5592404,9786708,22369620,
%U 39146836,89478484,156587348,357913940,626349396,1431655764,2505397588
%N Expansion of (1+3x)/((1-x)(1-4x^2)).
%C Partial sums of A084221. a(n) = A097163(n+1)/4. Third binomial transform is A097165.
%C a(n+1) = 4*A097163(n). - _Zerinvary Lajos_, Mar 17 2008
%C See A133628 for an essentially identical sequence. - _R. J. Mathar_, Jun 08 2008
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-4).
%F a(n) = 5*2^n/2 - (-2)^n/6 - 4/3;
%F a(n) = a(n-1) + 4a(n-2) - 4a(n-3).
%F G.f. ( 1+3*x ) / ( (x-1)*(2*x+1)*(2*x-1) ). - _R. J. Mathar_, Jul 06 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], n=1..31); # _Zerinvary Lajos_, Mar 17 2008
%t CoefficientList[Series[(1+3x)/((1-x)(1-4x^2)),{x,0,50}],x] (* or *) LinearRecurrence[{1,4,-4},{1,4,8},50] (* _Harvey P. Dale_, Jul 11 2023 *)
%K easy,nonn
%O 0,2
%A _Paul Barry_, Jul 30 2004
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