This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A178420 #71 Dec 22 2023 10:36:08
%S A178420 0,1,3,8,18,39,81,166,336,677,1359,2724,5454,10915,21837,43682,87372,
%T A178420 174753,349515,699040,1398090,2796191,5592393,11184798,22369608,
%U A178420 44739229,89478471,178956956,357913926,715827867,1431655749,2863311514
%N A178420 Partial sums of floor(2^n/3).
%C A178420 Essentially the same as A011377: 0 followed by the terms of A011377. - _Joerg Arndt_, Apr 22 2016
%C A178420 Partial sums of A000975(n-1).
%H A178420 Vincenzo Librandi, <a href="/A178420/b178420.txt">Table of n, a(n) for n = 1..1000</a>
%H A178420 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 A178420 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-3,2).
%F A178420 a(n) = A011377(n-1) for n >= 1. - _Joerg Arndt_, Apr 22 2016
%F A178420 a(n) = round((8*2^n - 6*n - 9)/12).
%F A178420 a(n) = floor((4*2^n - 3*n - 4)/6).
%F A178420 a(n) = ceiling((4*2^n - 3*n - 5)/6).
%F A178420 a(n) = round((4*2^n - 3*n - 4)/6).
%F A178420 a(n) = a(n-2) + 2^(n-1) - 1, n > 2.
%F A178420 From _Bruno Berselli_, Jan 15 2011: (Start)
%F A178420 a(n) = (8*2^n - 6*n - 9 + (-1)^n)/12.
%F A178420 G.f.: x^2/((1+x)*(1-2*x)*(1-x)^2). (End)
%F A178420 G.f.: Q(0)/(3*(1-x)^2), where Q(k) = 1 - 1/(4^k - 2*x*16^k/(2*x*4^k - 1/(1 + 1/(2*4^k - 8*x*16^k/(4*x*4^k + 1/Q(k+1)))))); (continued fraction). - _Sergei N. Gladkovskii_, May 21 2013
%F A178420 a(n) = 2*a(n-1) + floor(n/2) for n > 1. - _Bruno Berselli_, Apr 30 2014
%F A178420 a(n) = floor(2^(n+1)/3) - floor((n+1)/2). - _Seiichi Manyama_, Dec 22 2023
%e A178420 a(5) = 0 + 1 + 2 + 5 + 10 = 18.
%p A178420 seq(round((4*2^n-3*n-4)/6),n=1..50)
%t A178420 f[n_] := Floor[(4 2^n - 3 n - 4)/6]; f[Range[60]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 29 2011 *)
%t A178420 CoefficientList[Series[x / ((1 + x) (1 - 2 x) (1 - x)^2), {x, 0, 50}], x] (* _Vincenzo Librandi_, Mar 26 2014 *)
%t A178420 LinearRecurrence[{3,-1,-3,2},{0,1,3,8},40] (* or *) Accumulate[ Table[ Floor[ 2^n/3],{n,40}]] (* _Harvey P. Dale_, Dec 24 2015 *)
%o A178420 (Magma) [Floor((4*2^n-3*n-4)/6): n in [1..30]]; // _Vincenzo Librandi_, Jun 23 2011
%o A178420 (PARI) a(n)=(4<<n-3*n-4)\6 \\ _Charles R Greathouse IV_, Jul 31 2013
%Y A178420 Column k=2 of A368296.
%Y A178420 Cf. A000975, A178455.
%K A178420 nonn,easy
%O A178420 1,3
%A A178420 _Mircea Merca_, Dec 21 2010