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A137402 a(n) = Sum_{k=0..n} binomial(floor(n-2k/3), k).

%I A137402 #23 Sep 08 2022 08:45:32
%S A137402 1,1,2,3,5,9,16,28,48,81,136,229,388,661,1129,1928,3287,5594,9510,
%T A137402 16164,27484,46757,79577,135454,230552,392355,667620,1135924,1932721,
%U A137402 3288563,5595805,9522067,16203273,27572144,46917243,79834375,135845607,231154212
%N A137402 a(n) = Sum_{k=0..n} binomial(floor(n-2k/3), k).
%C A137402 A_n + B_{n-1} + C_{n-2} in the notation of A137356.
%C A137402 Lim_{n->infinity} a(n+1)/a(n) = x ~= 1.7016..., with x given by the real root (A324498) of (x - 1)^3*x^2 = 1. - _Hugo Pfoertner_, Mar 15 2019
%D A137402 D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.
%H A137402 Colin Barker, <a href="/A137402/b137402.txt">Table of n, a(n) for n = 0..1000</a>
%H A137402 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,0,1).
%F A137402 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-5); a(0)=1, a(1)=1, a(2)=2, a(3)=3, a(4)=5. - _Harvey P. Dale_, Aug 22 2011
%F A137402 G.f.: (1 - 2*x + 2*x^2 - x^3 + x^4) / (1 - 3*x + 3*x^2 - x^3 - x^5). - _Colin Barker_, Dec 14 2015
%p A137402 f:=n->add( binomial( floor(n-2*k/3), k), k=0..n);
%t A137402 Table[Sum[Binomial[Floor[n-(2k)/3],k],{k,0,n}],{n,0,40}] (* or *) LinearRecurrence[{3,-3,1,0,1},{1,1,2,3,5},40] (* _Harvey P. Dale_, Aug 22 2011 *)
%o A137402 (PARI) Vec((1-2*x+2*x^2-x^3+x^4)/(1-3*x+3*x^2-x^3-x^5) + O(x^50)) \\ _Colin Barker_, Dec 14 2015
%o A137402 (PARI) a(n) = sum(k=0, n, binomial(floor(n-2*k/3), k)); \\ _Altug Alkan_, Dec 14 2015
%o A137402 (Magma) [(&+[Binomial(Floor(n-2*k/3), k): k in [0..n]]): n in [0..40]]; // _G. C. Greubel_, Mar 15 2019
%o A137402 (Sage) [sum(binomial(floor(n-2*k/3),k) for k in (0..n)) for n in (0..40)] # _G. C. Greubel_, Mar 15 2019
%Y A137402 Cf. A137356, A137357, A137358, A324498.
%K A137402 nonn,easy
%O A137402 0,3
%A A137402 _Don Knuth_, Apr 11 2008