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A178459 Partial sums of floor(2^n/31).

%I A178459 #34 May 06 2024 06:52:04
%S A178459 0,0,0,0,1,3,7,15,31,64,130,262,526,1054,2111,4225,8453,16909,33821,
%T A178459 67646,135296,270596,541196,1082396,2164797,4329599,8659203,17318411,
%U A178459 34636827,69273660,138547326,277094658,554189322,1108378650,2216757307,4433514621,8867029249,17734058505,35468117017,70936234042
%N A178459 Partial sums of floor(2^n/31).
%C A178459 Partial sums of A119610(n-4).
%H A178459 Vincenzo Librandi, <a href="/A178459/b178459.txt">Table of n, a(n) for n = 1..1000</a>
%H A178459 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 A178459 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,0,0,1,-3,2).
%F A178459 a(n) = round((10*2^n - 31*n - 7)/155).
%F A178459 a(n) = floor((10*2^n - 31*n + 22)/155).
%F A178459 a(n) = ceiling((10*2^n - 31*n - 36)/155).
%F A178459 a(n) = round((10*2^n - 31*n - 10)/155).
%F A178459 a(n) = a(n-5) + 2^(n-4) - 1, n > 4,
%F A178459 G.f.: -x^5/((x-1)^2*(2*x-1)*(x^4 + x^3 + x^2 + x + 1)). - _Colin Barker_, Oct 27 2012
%F A178459 From _Seiichi Manyama_, Dec 22 2023: (Start)
%F A178459 a(n) = Sum_{k=0..n} 2^(n-k) * floor(k/5).
%F A178459 a(n) = floor(2^(n+1)/31) - floor((n+1)/5). (End)
%e A178459 a(29) = 0 + 0 + 0 + 0 + 1 + 2 + 4 + 8 + 16 + 33 + 66 + 132 + 264 + 528 + 1057 + 2114 + 4228 + 8456 + 16912 + 33825 + 67650 + 135300 + 270600 + 541200 + 1082401 + 2164802 + 4329604 + 8659208 + 17318416 = 34636827.
%p A178459 seq(round((10*2^n-31*n-7)/155),n=1..32)
%t A178459 Floor[2^Range[40]/31]//Accumulate (* _Harvey P. Dale_, May 11 2018 *)
%o A178459 (Magma) [Round((10*2^n-31*n-7)/155): n in [1..40]]; // _Vincenzo Librandi_, Jun 21 2011
%Y A178459 Cf. A119610.
%K A178459 nonn,easy
%O A178459 1,6
%A A178459 _Mircea Merca_, Dec 22 2010