%I #40 Sep 08 2022 08:45:54
%S 0,0,1,5,17,52,156,468,1405,4217,12653,37960,113880,341640,1024921,
%T 3074765,9224297,27672892,83018676,249056028,747168085,2241504257,
%U 6724512773,20173538320,60520614960,181561844880
%N Partial sums of round(3^n/7).
%H Vincenzo Librandi, <a href="/A178703/b178703.txt">Table of n, a(n) for n = 0..500</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 (5,-8,7,-3).
%F a(n) = round((3*3^n - 7)/14).
%F a(n) = floor((3*3^n - 1)/14).
%F a(n) = ceiling((3*3^n - 13)/14).
%F a(n) = a(n-6) + 52*3^(n-5), n > 5.
%F a(n) = 5*a(n-1) - 8*a(n-2) + 7*a(n-3) - 3*a(n-4), n > 3.
%F G.f.: x^2/((1 - x)*(1 - 3*x)*(1 - x + x^2)).
%F a(n) = 3^(n+1)/14 - 1/2 + A174737(n)/7. - _R. J. Mathar_, Jan 08 2011
%e a(6) = 0 + 0 + 1 + 4 + 12 + 35 + 104 = 156.
%p A178703 := proc(n) add( round(3^i/7),i=0..n) ; end proc:
%t Table[Floor[(3^(n+1)-1)/14], {n,0,30}] (* _G. C. Greubel_, Jan 25 2019 *)
%o (Magma) [Floor((3*3^n-1)/14): n in [0..30]]; // _Vincenzo Librandi_, May 01 2011
%o (PARI) vector(30, n, n--; ((3^(n+1)-1)/14)\1) \\ _G. C. Greubel_, Jan 25 2019
%o (Sage) [floor((3^(n+1)-1)/14) for n in (0..30)] # _G. C. Greubel_, Jan 25 2019
%K nonn,less,easy
%O 0,4
%A _Mircea Merca_, Dec 28 2010