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
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).
Programs
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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
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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
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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 ).
Extensions
More terms from Vincenzo Librandi, Mar 01 2014