%I #25 Jul 04 2019 03:40:03
%S 0,1,2,3,4,6,8,12,18,27,41,64,100,159,254,407,654,1054,1700,2746,4438,
%T 7175,11603,18768,30360,49117,79466,128571,208024,336582,544592,
%U 881160,1425738,2306883,3732605,6039472,9772060
%N Partial sums of ceiling(Fibonacci(n)/4).
%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_09">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,0,0,1,-2,0,1).
%F a(n) = round(Fibonacci(n+2)/4 + n/2).
%F a(n) = floor(Fibonacci(n+2)/4 + n/2 + 1/4).
%F a(n) = ceiling(Fibonacci(n+2)/4 + n/2 - 1/4).
%F a(n) = round(Fibonacci(n+2)/4 + n/2 - 1/4).
%F a(n) = a(n-6) + Fibonacci(n-1) + 3, n > 5.
%F a(n) = 2*a(n-1) - a(n-3) + a(n-6) - 2*a(n-7) + a(n-9), n > 8.
%F G.f.: (x^7 + x^6 + x^4 + x^3 - x)/((x+1)*(x^2 + x + 1)*(x^2 - x + 1)*(x^2 + x - 1)*(x-1)^2).
%e a(7) = 0 + 1 + 1 + 1 + 1 + 2 + 2 + 4 = 12.
%p A179042 := proc(n) add( ceil(combinat[fibonacci](i)/4),i=0..n) ; end proc:
%t Accumulate[Ceiling[Fibonacci[Range[0,40]]/4]] (* or *) LinearRecurrence[ {2,0,-1,0,0,1,-2,0,1},{0,1,2,3,4,6,8,12,18},40] (* _Harvey P. Dale_, May 25 2014 *)
%K nonn
%O 0,3
%A _Mircea Merca_, Jan 04 2011