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%I #19 Sep 22 2023 11:05:07
%S 1,7,22,70,211,637,1912,5740,17221,51667,155002,465010,1395031,
%T 4185097,12555292,37665880,112997641,338992927,1016978782,3050936350,
%U 9152809051,27458427157,82375281472,247125844420,741377533261,2224132599787,6672397799362,20017193398090
%N Expansion of g.f. (1+4*x)/((1-x)*(1+x)*(1-3*x)).
%C Binomial transform of A084431.
%H Vincenzo Librandi, <a href="/A085287/b085287.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-3).
%F a(n) = (-10 - 3(-1)^n + 21*3^n)/8.
%F a(n) = 2*a(n-1) + 3*a(n-2) + 5, a(0)=0, a(1)=1. - _Zerinvary Lajos_, Dec 14 2008
%F From _Stefano Spezia_, Sep 20 2023: (Start)
%F a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3) for n > 2.
%F E.g.f.: exp(x)*(9*cosh(2*x) + 12*sinh(2*x) - 5)/4. (End)
%p a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2]+5 od: seq(a[n], n=1..33); # _Zerinvary Lajos_, Dec 14 2008
%t LinearRecurrence[{3,1,-3},{1,7,22},30] (* _Harvey P. Dale_, Sep 22 2023 *)
%o (Magma) [(-10-3*(-1)^n+21*3^n)/8: n in [0..30]]; // _Vincenzo Librandi_, Nov 16 2011
%o (PARI) Vec((1+4*x)/((1-x^2)*(1-3*x)) + O(x^30)) \\ _Michel Marcus_, Aug 14 2017
%Y Cf. A084431.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Jun 26 2003
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