A179001 - OEIS

%I #30 Sep 08 2022 08:45:54

%S 0,0,0,0,1,2,4,8,15,26,44,73,121,198,323,526,855,1387,2248,3641,5896,

%T 9544,15447,24999,40455,65463,105927,171399,277336,448745,726091,

%U 1174847,1900950,3075809,4976771,8052592,13029376,21081981,34111370,55193365,89304750

%N Partial sums of floor(Fibonacci(n)/3).

%C Partial sums of A004696.

%H Vincenzo Librandi, <a href="/A179001/b179001.txt">Table of n, a(n) for n = 0..280</a>

%H Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Merca/merca3.html">Inequalities and Identities Involving Sums of Integer Functions</a> J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.

%F a(n) = round(Fibonacci(n+2)/3 - 3*n/8 - 11/24).

%F a(n) = round(Fibonacci(n+2)/3 - 3*n/8 - 1/3).

%F a(n) = floor(Fibonacci(n+2)/3 - 3*n/8 - 1/6).

%F a(n) = ceiling(Fibonacci(n+2)/3 - 3*n/8 - 3/4).

%F a(n) = a(n-8) + Fibonacci(n-1) + Fibonacci(n-3) - 3, n > 8.

%F a(n) = 2*a(n-1) - a(n-3) + a(n-8) - 2*a(n-9) + a(n-11), n > 10.

%F G.f.: -x^4*(1 + x^4 + x^3) / ( (1+x)*(x^2+1)*(x^2+x-1)*(x^4+1)*(x-1)^2 ).

%e a(9) = 0 + 0 + 0 + 0 + 1 + 1 + 2 + 4 + 7 + 11 = 26.

%p A179001 := proc(n) add( floor(combinat[fibonacci](i)/3),i=0..n) ; end proc:

%t Accumulate[Floor[Fibonacci[Range[0,40]]/3]] (* _Harvey P. Dale_, Jun 13 2022 *)

%o (Magma) [Floor(Fibonacci(n+2)/3-3*n/8-1/6): n in [0..40]]; // _Vincenzo Librandi_, Apr 28 2011

%Y Cf. A004696.

%K nonn

%O 0,6

%A _Mircea Merca_, Jan 03 2011