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a=b=c=0;c(n)=c+n+a;b(n)=b+n+c;a(n)=a+n+b.
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%I #6 Jun 13 2015 00:52:45

%S 0,3,15,55,183,588,1865,5887,18550,58414,183904,578936,1822459,

%T 5736946,18059358,56849069,178955165,563332829,1773314909,5582216334,

%U 17572253459,55315679765,174128175040,548137914348,1725482812062

%N a=b=c=0;c(n)=c+n+a;b(n)=b+n+c;a(n)=a+n+b.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (6, -12, 11, -5, 1).

%F G.f.: x^2(3-3x+x^2)/((1-x)^2*(1-4x+3x^2-x^3)). a(n)=6*a(n-1)-12*a(n-2)+11*a(n-3)-5*a(n-4)+a(n-5). [From _R. J. Mathar_, Dec 15 2008]

%t lst={};a=b=c=0;Do[c+=n+a;b+=n+c;a+=n+b;AppendTo[lst,a],{n,0,2*4!}];lst

%t LinearRecurrence[{6,-12,11,-5,1},{0,3,15,55,183},30] (* _Harvey P. Dale_, Dec 11 2014 *)

%Y Cf. A152891, A152895

%K nonn

%O 1,2

%A _Vladimir Joseph Stephan Orlovsky_, Dec 14 2008