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%I #40 Sep 08 2022 08:45:54
%S 0,0,0,0,1,4,11,25,53,109,222,449,904,1814,3634,7274,14555,29118,
%T 58245,116499,233007,466023,932056,1864123,3728258,7456528,14913068,
%U 29826148,59652309,119304632,238609279,477218573,954437161
%N Partial sums of floor(2^n/9).
%C Partial sums of A153234.
%H Vincenzo Librandi, <a href="/A178742/b178742.txt">Table of n, a(n) for n = 0..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_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,1,4,-5,2).
%F a(n) = round((8*2^n - 18*n - 9)/36).
%F a(n) = floor((4*2^n - 9*n + 2)/18).
%F a(n) = ceiling((4*2^n - 9*n - 11)/18).
%F a(n) = round((4*2^n - 9*n - 4)/18).
%F a(n) = a(n-6) + 7*2^(n-5) - 3, n > 5.
%F a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 4*a(n-4) - 5*a(n-5) + 2*a(n-6).
%F G.f.: x^4 / ( (1-2*x)*(1+x)*(1-x+x^2)*(1-x)^2 ).
%e a(6) = 0 + 0 + 0 + 0 + 1 + 3 + 7 = 11.
%p A178742 := proc(n) add( floor(2^i/9),i=0..n) ; end proc:
%t CoefficientList[Series[x^4/((1-2x)(1+x)(1-x+x^2)(1-x)^2), {x, 0, 50}], x] (* _Vincenzo Librandi_, Mar 26 2014 *)
%t LinearRecurrence[{4,-5,1,4,-5,2},{0,0,0,0,1,4},40] (* _Harvey P. Dale_, Jan 25 2015 *)
%o (Magma) [&+[Floor(2^k/9): k in [0..n]]: n in [0..25]]; // _Bruno Berselli_, Apr 26 2011
%o (Magma) I:=[0,0,0,0,1,4]; [n le 6 select I[n] else 4*Self(n-1)-5*Self(n-2)+Self(n-3)+4*Self(n-4)-5*Self(n-5)+2*Self(n-6): n in [1..40]]; // _Vincenzo Librandi_, Mar 26 2014
%o (PARI) vector(30, n, n--; ((4*2^n-9*n+2)/18)\1) \\ _G. C. Greubel_, Jan 24 2019
%o (Sage) [floor((4*2^n-9*n+2)/18) for n in (0..30)] # _G. C. Greubel_, Jan 24 2019
%Y Cf. A153234.
%K nonn,less
%O 0,6
%A _Mircea Merca_, Dec 26 2010
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