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a(n) = C(n-2,2)+C(n-5,5)+...+C(n-(3*floor((n-3)/3)+2),3*floor((n-3)/3)+2).
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%I #16 Dec 30 2023 22:05:37

%S 0,0,0,0,1,3,6,10,15,21,29,42,66,111,192,330,554,906,1452,2303,3651,

%T 5826,9382,15225,24807,40431,65748,106584,172321,278184,448980,725140,

%U 1172412,1897380,3072365,4975551,8055918,13038606,21096027,34125561

%N a(n) = C(n-2,2)+C(n-5,5)+...+C(n-(3*floor((n-3)/3)+2),3*floor((n-3)/3)+2).

%H Vincenzo Librandi, <a href="/A101551/b101551.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,0,0,1)

%F G.f.: x^4/((1-x)^3-x^6) = -x^4/ ((x^2+x-1)*(x^4-x^3+2*x^2-2*x+1)).

%F a(n) = Sum_{k=0..n} if(mod(k+1, 3)=0, C(n-k, k), 0).

%F a(n+2) = Sum_{k=0..floor(n/6)} binomial(n-3k, 3k+2). - _Paul Barry_, Jan 13 2005

%t CoefficientList[Series[x^4/((1-x)^3-x^6),{x,0,50}],x] (* _Vincenzo Librandi_, Jul 08 2012 *)

%t LinearRecurrence[{3,-3,1,0,0,1},{0,0,0,0,1,3},40] (* _Harvey P. Dale_, Feb 20 2014 *)

%Y Cf. A024490, A101552.

%K easy,nonn

%O 0,6

%A _Paul Barry_, Dec 06 2004