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%I #30 Sep 26 2023 02:09:08
%S 1,0,3,3,9,18,36,81,162,351,729,1539,3240,6804,14337,30132,63423,
%T 133407,280665,590490,1242216,2613465,5498118,11567043,24334749,
%U 51195483,107705376,226590696,476702577,1002888216,2109879819,4438772379
%N a(n) = 3*a(n-2) + 3*a(n-3), a(0)=1, a(1)=0, a(2)=3.
%C Diagonal sums of A099093.
%C Counts walks (closed) on the graph G(1-vertex; 2-loop, 2-loop, 2-loop, 3-loop, 3-loop, 3-loop). - _David Neil McGrath_, Jan 16 2015
%C Number of compositions of n into parts 2 and 3, each of three sorts. - _Joerg Arndt_, Feb 14 2015
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,3).
%F G.f.: 1/(1 - 3*x^2 - 3*x^3).
%F a(n) = Sum_{k=0..floor(n/2)} binomial(k, n-2k)*3^k.
%F Construct the power matrix T(n,j) = [A(n)^*j]*[S(n)^*(j-1)] where A(n) = (0,3,3,0,0,...) and S(n) = (0,1,0,0,...). (* is convolution operation.) Define S^*0 = I. Then T(n,j) counts n-walks containing (j) loops, on the single vertex graph above, and a(n) = Sum_{j=1..n} T(n,j). - _David Neil McGrath_, Jan 16 2015
%t CoefficientList[Series[1 / (1 - 3 x^2 - 3 x^3), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 16 2015 *)
%t LinearRecurrence[{0,3,3},{1,0,3},40] (* _Harvey P. Dale_, Aug 15 2017 *)
%o (PARI) Vec(1/(1-3*x^2-3*x^3) + O(x^50)) \\ _Michel Marcus_, Jan 17 2015
%K easy,nonn
%O 0,3
%A _Paul Barry_, Sep 25 2004
%E Corrected by _Philippe Deléham_, Dec 18 2008
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