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

1, 1, 3, 5, 10, 16, 29, 47, 81, 131, 220, 356, 589, 953, 1563, 2529, 4126, 6676, 10857, 17567, 28513, 46135, 74792, 121016, 196041, 317201, 513619, 831053, 1345282, 2176712, 3522981, 5700303, 9224881, 14926171, 24153636, 39081404, 63239221, 102323209

Offset: 0

Author

Paul Barry, Apr 11 2007

Keywords

• nonn
• easy

Examples

			  1 =  1.
  1 =  1.
  1 +  2 =  3.
  2 +  3 =  5.
  2 +  3 +  5 = 10.
  3 +  5 +  8 = 16.
  3 +  5 +  8 + 13 = 29.
  5 +  8 + 13 + 21 = 47.
  5 +  8 + 13 + 21 + 34 =  81.
  8 + 13 + 21 + 34 + 55 = 131.
  8 + 13 + 21 + 34 + 55 +  89 = 220.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-1,0,-1,-1).

Crossrefs

Cf. A000045, A103609, A129362.

Programs

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

Mathematica

a[n_]:= Sum[Fibonacci@k, {k, Floor[(n + 3)/2], n + 1}]; Array[a, 33, 0] (* Robert G. Wilson v, Mar 15 2011 *)
Table[Sum[Fibonacci[n - i + 2], {i, Floor[(n + 2)/2]}], {n, 0, 50}] (* Wesley Ivan Hurt, Feb 25 2014 *)
LinearRecurrence[{1,2,-1,0,-1,-1},{1,1,3,5,10,16},40] (* Harvey P. Dale, Feb 02 2019 *)

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

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

Formula

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

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

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

a(n) = A000045(n+3) - A103609(n+5). - R. J. Mathar, Mar 15 2011

Extensions

More terms from Vincenzo Librandi, Mar 01 2014