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%I #34 May 01 2020 17:35:21
%S 0,0,0,0,0,1,3,6,11,19,32,54,90,148,242,394,640,1039,1685,2730,4421,
%T 7157,11584,18748,30340,49096,79444,128548,208000,336557,544567,
%U 881134,1425711,2306855,3732576,6039442,9772030,15811484
%N Partial sums of floor(Fibonacci(n)/4).
%C Partial sums of A004697.
%H Andrew Howroyd, <a href="/A179006/b179006.txt">Table of n, a(n) for n = 0..1000</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.
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2,-2,2,-1,-1,1).
%F a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - a(n-6) - a(n-7) + a(n-8).
%F a(n) = round(Fibonacci(n+2)/4 - n/3 - 3/8).
%F a(n) = round(Fibonacci(n+2)/4 - n/3 - 1/4).
%F a(n) = floor(Fibonacci(n+2)/4 - n/3 - 1/12).
%F a(n) = ceiling(Fibonacci(n+2)/4 - n/3 - 2/3).
%F a(n) = a(n-6) + Fibonacci(n-1) - 2, n > 6.
%F G.f.: -x^5/((x^2+x+1)*(x^2-x+1)*(x^2+x-1)*(x-1)^2).
%e a(7) = 0 + 0 + 0 + 0 + 0 + 1 + 2 + 3 = 6.
%p A179006 := proc(n) add( floor(combinat[fibonacci](i)/4),i=0..n) ; end proc:
%t f[n_] := Floor[ Fibonacci@ n/4]; Accumulate@ Array[f, 38]
%t LinearRecurrence[{3,-3,2,-2,2,-1,-1,1},{0,0,0,0,0,1,3,6},40] (* _Harvey P. Dale_, Jan 28 2020 *)
%o (PARI) a(n)={round(fibonacci(n+2)/4 - n/3 - 3/8)} \\ _Andrew Howroyd_, May 01 2020
%Y Cf. A004697.
%K nonn
%O 0,7
%A _Mircea Merca_, Jan 03 2011
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