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%I #24 Apr 08 2024 12:47:53
%S 1,2,3,5,10,22,50,115,266,617,1433,3330,7740,17992,41825,97230,226031,
%T 525457,1221538,2839730,6601570,15346787,35676950,82938845,192809421,
%U 448227522,1042002568,2422362080,5631308625,13091204282,30433357675,70748973085,164471408186
%N Sum of every third term of the Padovan sequence A000931.
%H Colin Barker, <a href="/A329244/b329244.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,3,-1).
%F a(n) = Sum_{i=0..n} A000931(3*i).
%F a(n) = A000931(3n+2)+1.
%F From _Colin Barker_, Nov 09 2019: (Start)
%F G.f.: (1 - 2*x) / ((1 - x)*(1 - 3*x + 2*x^2 - x^3)).
%F a(n) = 4*a(n-1) - 5*a(n-2) + 3*a(n-3) - a(n-4) for n>3. (End)
%e For n = 3, a(3) = 1+1+1+2 = 5.
%t LinearRecurrence[{4, -5, 3, -1}, {1, 2, 3, 5}, 50] (* _Paolo Xausa_, Apr 08 2024 *)
%o (Python)
%o p = lambda x:[1, 0, 0][x] if x<3 else p(x-2)+p(x-3)
%o a = lambda x:sum(p(3*i) for i in range(x+1))
%o (PARI) Vec((1 - 2*x) / ((1 - x)*(1 - 3*x + 2*x^2 - x^3)) + O(x^35)) \\ _Colin Barker_, Nov 09 2019
%Y Partial sums of A034943.
%Y Cf. A000931.
%K nonn,easy,changed
%O 0,2
%A _David Nacin_, Nov 09 2019
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