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%I #15 Sep 09 2017 03:22:29
%S 1,2,4,9,23,64,186,551,1645,4926,14768,44293,132867,398588,1195750,
%T 3587235,10761689,32285050,96855132,290565377,871696111,2615088312,
%U 7845264914,23535794719,70607384133,211822152374,635466457096
%N a(n+1) = 3*a(n) - n.
%H G. C. Greubel, <a href="/A164039/b164039.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 (5,-7,3).
%F a(0) = 1; a(n+1) = 3*a(n) - n.
%F a(n) = (3^n + 2*n + 3)/4.
%F From _R. J. Mathar_, Aug 09 2009: (Start)
%F a(n) = 5*a(n-1)-7*a(n-2)+3*a(n-3).
%F G.f.: (1-3*x+x^2)/((1-3*x)*(1-x)^2). (End)
%F E.g.f.: (1/4)*((2*x + 3)*exp(x) + exp(3*x)). - _G. C. Greubel_, Sep 08 2017
%t Transpose[NestList[Flatten[{Rest[#],ListCorrelate[#,{3,-7,5}]}]&, {1,2,4},30]][[1]] (* _Harvey P. Dale_, Mar 24 2011 *)
%t Table[(3^n + 2*n + 3)/4, {n,0,50}] (* _G. C. Greubel_, Sep 08 2017 *)
%o (PARI) x='x+O('x^50); Vec((1-3*x+x^2)/((1-3*x)*(1-x)^2)) \\ _G. C. Greubel_, Sep 08 2017
%K nonn,easy
%O 0,2
%A _Rolf Pleisch_, Aug 08 2009
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