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A129361 a(n) = Sum_{k=floor((n+1)/2)..n} Fibonacci(k+1).

%I A129361 #38 Feb 01 2024 01:49:33
%S A129361 1,1,3,5,10,16,29,47,81,131,220,356,589,953,1563,2529,4126,6676,10857,
%T A129361 17567,28513,46135,74792,121016,196041,317201,513619,831053,1345282,
%U A129361 2176712,3522981,5700303,9224881,14926171,24153636,39081404,63239221,102323209
%N A129361 a(n) = Sum_{k=floor((n+1)/2)..n} Fibonacci(k+1).
%H A129361 Vincenzo Librandi, <a href="/A129361/b129361.txt">Table of n, a(n) for n = 0..1000</a>
%H A129361 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1,0,-1,-1).
%F A129361 G.f.: (1+x)*(1-x+x^2)/((1-x-x^2)*(1-x^2-x^4)).
%F A129361 a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-5) - a(n-6).
%F A129361 a(n) = Sum_{k=0..n} ( F(k+1) - F((k+1)/2)*(1-(-1)^k)/2 ).
%F A129361 a(n) = A000045(n+3) - A103609(n+5). - _R. J. Mathar_, Mar 15 2011
%e A129361   1 =  1.
%e A129361   1 =  1.
%e A129361   1 +  2 =  3.
%e A129361   2 +  3 =  5.
%e A129361   2 +  3 +  5 = 10.
%e A129361   3 +  5 +  8 = 16.
%e A129361   3 +  5 +  8 + 13 = 29.
%e A129361   5 +  8 + 13 + 21 = 47.
%e A129361   5 +  8 + 13 + 21 + 34 =  81.
%e A129361   8 + 13 + 21 + 34 + 55 = 131.
%e A129361   8 + 13 + 21 + 34 + 55 +  89 = 220.
%t A129361 a[n_]:= Sum[Fibonacci@k, {k, Floor[(n + 3)/2], n + 1}]; Array[a, 33, 0] (* _Robert G. Wilson v_, Mar 15 2011 *)
%t A129361 Table[Sum[Fibonacci[n - i + 2], {i, Floor[(n + 2)/2]}], {n, 0, 50}] (* _Wesley Ivan Hurt_, Feb 25 2014 *)
%t A129361 LinearRecurrence[{1,2,-1,0,-1,-1},{1,1,3,5,10,16},40] (* _Harvey P. Dale_, Feb 02 2019 *)
%o A129361 (Magma) I:=[1,1,3,5,10,16]; [n le 6 select I[n] else Self(n-1) +2*Self(n-2)-Self(n-3)-Self(n-5)-Self(n-6): n in [1..50]]; // _Vincenzo Librandi_, Mar 01 2014
%o A129361 (PARI) Vec( (1+x)*(1-x+x^2)/((1-x-x^2)*(1-x^2-x^4)) +O(x^66) ) \\ _Joerg Arndt_, Mar 01 2014
%o A129361 (SageMath) [sum(fibonacci(n-j+2) for j in range(1,2+(n//2))) for n in range(51)] # _G. C. Greubel_, Jan 31 2024
%Y A129361 Cf. A000045, A103609, A129362.
%K A129361 nonn,easy
%O A129361 0,3
%A A129361 _Paul Barry_, Apr 11 2007
%E A129361 More terms from _Vincenzo Librandi_, Mar 01 2014