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A129362 a(n) = Sum_{k=floor((n+1)/2)..n} J(k+1), J(k) = A001045(k).

%I A129362 #17 Feb 01 2024 01:52:02
%S A129362 1,1,4,8,19,37,80,160,331,661,1344,2688,5419,10837,21760,43520,87211,
%T A129362 174421,349184,698368,1397419,2794837,5591040,11182080,22366891,
%U A129362 44733781,89473024,178946048,357903019,715806037
%N A129362 a(n) = Sum_{k=floor((n+1)/2)..n} J(k+1), J(k) = A001045(k).
%H A129362 Harvey P. Dale, <a href="/A129362/b129362.txt">Table of n, a(n) for n = 0..1000</a>
%H A129362 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-1,0,-2,-4).
%F A129362 G.f.: (1+2*x^3)/((1-x-2*x^2)*(1-x^2-2*x^4)).
%F A129362 a(n) = a(n-1) + 3*a(n-2) - a(n-3) - 2*a(n-5) - 4*a(n-6).
%F A129362 a(n) = Sum_{k=0..n} ( J(k+1) - J((k+1)/2)*(1-(-1)^k)/2 ).
%F A129362 a(n) = Sum_{j=0..floor(n/2)} A001045(n-j+1). - _G. C. Greubel_, Jan 31 2024
%t A129362 LinearRecurrence[{1,3,-1,0,-2,-4},{1,1,4,8,19,37},30] (* _Harvey P. Dale_, Oct 22 2011 *)
%o A129362 (Magma)
%o A129362 A001045:= func< n | (2^n - (-1)^n)/3 >;
%o A129362 [(&+[A001045(n-j+1): j in [0..Floor(n/2)]]): n in [0..30]]; // _G. C. Greubel_, Jan 31 2024
%o A129362 (SageMath)
%o A129362 def A001045(n): return (2^n - (-1)^n)/3
%o A129362 def A129362(n): return sum(A001045(n-j+1) for j in range(1+(n//2)))
%o A129362 [A129362(n) for n in range(31)] # _G. C. Greubel_, Jan 31 2024
%Y A129362 Cf. A001045, A129361.
%K A129362 nonn,easy
%O A129362 0,3
%A A129362 _Paul Barry_, Apr 11 2007