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A026647 a(n) = Sum_{k=0..floor(n/2)} A026637(n-k, k).

%I A026647 #14 Jan 19 2025 15:08:12
%S A026647 1,1,2,3,6,10,17,27,45,73,119,192,312,505,818,1323,2142,3466,5609,
%T A026647 9075,14685,23761,38447,62208,100656,162865,263522,426387,689910,
%U A026647 1116298,1806209,2922507,4728717,7651225,12379943,20031168
%N A026647 a(n) = Sum_{k=0..floor(n/2)} A026637(n-k, k).
%H A026647 G. C. Greubel, <a href="/A026647/b026647.txt">Table of n, a(n) for n = 0..1000</a>
%H A026647 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,1,-1,-1).
%F A026647 G.f.: (1 + x^5 + x^6)/((1-x^4)*(1-x-x^2)).
%F A026647 From _G. C. Greubel_, Jul 01 2024: (Start)
%F A026647 a(n) = [n=0] - (3/4) + (1/4)*(-1)^n - (1/10)*2^((1-(-1)^n)/2)*(-1)^floor((n+1)/2) + (3/5)*LucasL(n+1).
%F A026647 a(n) = (1/20)*( 12*LucasL(n+1) + 5*(-1)^n - 15 - 2*cos(n*Pi/2) + 4*sin(n*Pi/2) ) + [n=0].
%F A026647 a(n) = a(n-2) + a(n-3) + 2*a(n-4) + a(n-5) + 3, with a(0) = a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 6, a(5) = 10. (End)
%t A026647 a[n_]:= a[n]= If[n<6, Binomial[n, Floor[n/2]], a[n-2] +a[n-3] +2*a[n- 4] +a[n-5] +3]; (* a = A026647 *)
%t A026647 Table[a[n], {n,0,40}] (* _G. C. Greubel_, Jul 01 2024 *)
%t A026647 LinearRecurrence[{1,1,0,1,-1,-1},{1,1,2,3,6,10,17},40] (* _Harvey P. Dale_, Jan 19 2025 *)
%o A026647 (Magma) [1] cat [n le 5 select Binomial(n, Floor(n/2)) else Self(n-2) +Self(n-3) +2*Self(n-4) +Self(n-5) +3: n in [1..40]]; // _G. C. Greubel_, Jul 01 2024
%o A026647 (SageMath)
%o A026647 @CachedFunction
%o A026647 def a(n): # a = A026647
%o A026647     if n<6: return binomial(n, n//2)
%o A026647     else: return a(n-2) + a(n-3) + 2*a(n-4) + a(n-5) + 3
%o A026647 [a(n) for n in range(41)] # _G. C. Greubel_, Jul 01 2024
%Y A026647 Cf. A026637, A026638, A026639, A026640, A026641, A026642, A026643.
%Y A026647 Cf. A026644, A026645, A026646, A026966, A026967, A026968, A026969.
%Y A026647 Cf. A026970.
%K A026647 nonn
%O A026647 0,3
%A A026647 _Clark Kimberling_