|
%I #36 May 06 2024 06:51:36
%S 0,0,0,0,0,0,1,3,7,15,31,63,127,256,514,1030,2062,4126,8254,16510,
%T 33023,66049,132101,264205,528413,1056829,2113661,4227326,8454656,
%U 16909316,33818636,67637276
%N Partial sums of floor(2^n/127).
%C Partials sums of A117302.
%H Vincenzo Librandi, <a href="/A178460/b178460.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_09">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,0,0,0,0,1,-3,2).
%F a(n) = round((14*2^n - 127*n + 75)/889).
%F a(n) = floor((14*2^n - 127*n + 284)/889).
%F a(n) = ceiling((14*2^n - 127*n - 134)/889).
%F a(n) = round((14*2^n - 127*n - 14)/889).
%F a(n) = a(n-7) + 2^(n-6) - 1, n > 6.
%F a(n) = 3*a(n-1) - 2*a(n-2) + a(n-7) - 3*a(n-8) + 2*a(n-9), n > 9.
%F G.f.: -x^7/((2*x-1)*(x-1)^2*(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)).
%F From _Seiichi Manyama_, Dec 22 2023: (Start)
%F a(n) = Sum_{k=0..n} 2^(n-k) * floor(k/7).
%F a(n) = floor(2^(n+1)/127) - floor((n+1)/7). (End)
%e a(10) = a(3) + 2^4 - 1 = 15.
%p A178460 := proc(n) add( floor(2^i/127),i=0..n) ; end proc:
%o (Magma) [Round((14*2^n-127*n+75)/889): n in [1..40]]; // _Vincenzo Librandi_, Jun 21 2011
%Y Cf. A117302.
%K nonn,easy,less
%O 1,8
%A _Mircea Merca_, Dec 22 2010
|
