A129361 - OEIS

%I #38 Feb 01 2024 01:49:33

%S 1,1,3,5,10,16,29,47,81,131,220,356,589,953,1563,2529,4126,6676,10857,

%T 17567,28513,46135,74792,121016,196041,317201,513619,831053,1345282,

%U 2176712,3522981,5700303,9224881,14926171,24153636,39081404,63239221,102323209

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

%H Vincenzo Librandi, <a href="/A129361/b129361.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1,0,-1,-1).

%F G.f.: (1+x)*(1-x+x^2)/((1-x-x^2)*(1-x^2-x^4)).

%F a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-5) - a(n-6).

%F a(n) = Sum_{k=0..n} ( F(k+1) - F((k+1)/2)*(1-(-1)^k)/2 ).

%F a(n) = A000045(n+3) - A103609(n+5). - _R. J. Mathar_, Mar 15 2011

%e   1 =  1.

%e   1 =  1.

%e   1 +  2 =  3.

%e   2 +  3 =  5.

%e   2 +  3 +  5 = 10.

%e   3 +  5 +  8 = 16.

%e   3 +  5 +  8 + 13 = 29.

%e   5 +  8 + 13 + 21 = 47.

%e   5 +  8 + 13 + 21 + 34 =  81.

%e   8 + 13 + 21 + 34 + 55 = 131.

%e   8 + 13 + 21 + 34 + 55 +  89 = 220.

%t a[n_]:= Sum[Fibonacci@k, {k, Floor[(n + 3)/2], n + 1}]; Array[a, 33, 0] (* _Robert G. Wilson v_, Mar 15 2011 *)

%t Table[Sum[Fibonacci[n - i + 2], {i, Floor[(n + 2)/2]}], {n, 0, 50}] (* _Wesley Ivan Hurt_, Feb 25 2014 *)

%t LinearRecurrence[{1,2,-1,0,-1,-1},{1,1,3,5,10,16},40] (* _Harvey P. Dale_, Feb 02 2019 *)

%o (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 (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 (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 Cf. A000045, A103609, A129362.

%K nonn,easy

%O 0,3

%A _Paul Barry_, Apr 11 2007

%E More terms from _Vincenzo Librandi_, Mar 01 2014