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Partial sums of round(4^n/9).
1

%I #27 Sep 08 2022 08:45:54

%S 0,0,2,9,37,151,606,2426,9708,38835,155343,621377,2485512,9942052,

%T 39768214,159072861,636291449,2545165803,10180663218,40722652878,

%U 162890611520,651562446087,2606249784355,10424999137429,41699996549724,166799986198904

%N Partial sums of round(4^n/9).

%H Vincenzo Librandi, <a href="/A178875/b178875.txt">Table of n, a(n) for n = 0..170</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_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4,1,-5,4).

%F a(n) = round((4*4^n - 3*n - 4)/27) = round((8*4^n - 6*n - 17)/54).

%F a(n) = floor((4*4^n - 3*n - 4)/27).

%F a(n) = ceiling((4*4^n - 3*n - 13)/27).

%F a(n) = a(n-3) + (7*4^(n-2) - 1)/3, n > 2.

%F a(n) = 5*a(n-1) - 4*a(n-2) + a(n-3) - 5*a(n-4) + 4*a(n-5), n > 4.

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

%e a(3)=0+0+2+7=9.

%p A178875 := proc(n) add( round(4^i/9),i=0..n) ; end proc:

%t Accumulate[Round[4^Range[0,30]/9]] (* _Harvey P. Dale_, Dec 16 2012 *)

%o (Magma) [Floor((4*4^n-3*n-4)/27): n in [0..40]]; // _Vincenzo Librandi_, Apr 28 2011

%K nonn,less

%O 0,3

%A _Mircea Merca_, Dec 28 2010