A178222 - OEIS

%I #44 Sep 08 2022 08:45:53

%S 0,2,8,28,88,270,816,2456,7376,22138,66424,199284,597864,1793606,

%T 5380832,16142512,48427552,145282674,435848040,1307544140,3922632440,

%U 11767897342,35303692048,105911076168,317733228528

%N Partial sums of floor(3^n/4).

%C Partial sums of A081251(n-1).

%H Vincenzo Librandi, <a href="/A178222/b178222.txt">Table of n, a(n) for n = 1..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_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-4,3).

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

%F a(n) = floor((3*3^n - 4*n - 3)/8).

%F a(n) = ceiling((3*3^n - 4*n - 5)/8).

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

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

%F From _Bruno Berselli_, Jan 14 2011: (Start)

%F a(n) = (3*3^n - 4*n - 4 + (-1)^n)/8.

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

%e a(3) = 0 + 2 + 6 = 8.

%p seq (round ((3*3^n-4*n-3)/8), n=1..25);

%t Accumulate[Floor[3^Range[30]/4]] (* _Harvey P. Dale_, Nov 04 2011 *)

%t CoefficientList[Series[2 x/((1 + x) (1 - 3 x) (1 - x)^2), {x, 0, 50}], x] (* _Vincenzo Librandi_, Mar 26 2014 *)

%o (Magma) [Floor((3*3^n-4*n-3)/8): n in [1..30]]; // _Vincenzo Librandi_, Jun 23 2011

%Y Cf. A081251.

%K nonn,easy

%O 1,2

%A _Mircea Merca_, Dec 26 2010