A175870 - OEIS

%I #25 Sep 08 2022 08:45:52

%S 0,1,2,3,4,6,8,11,14,18,23,29,35,43,52,62,73,86,100,116,133,152,173,

%T 196,220,247,276,307,340,376,414,455,498,544,593,645,699,757,818,882,

%U 949,1020,1094,1172,1253,1338,1427,1520,1616,1717,1822

%N Partial sums of ceiling(n^2/24).

%C There are several sequences of integers of the form ceiling(n^2/k) for whose partial sums we can establish identities as following (only for k = 2,...,8,10,11,12, 14,15,16,19,20,23,24).

%H Vincenzo Librandi, <a href="/A175870/b175870.txt">Table of n, a(n) for n = 0..900</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 (2,-1,1,-1,-1,1,-1,2,-1).

%F a(n) = round((2*n+1)*(2*n^2 + 2*n + 95)/288).

%F a(n) = floor((n+1)*(2*n^2 + n + 95)/144).

%F a(n) = ceiling((2*n^3 + 3*n^2 + 96*n)/144).

%F a(n) = a(n-24) + (n+1)*(n-24) + 220.

%F G.f.: x*(x^6 - x^3 + 1)/((x-1)^4*(x+1)*(x^2+1)*(x^2+x+1)). - _Colin Barker_, Oct 26 2012

%e a(24) = 0 + 1 + 1 + 1 + 1 + 2 + 2 + 3 + 3 + 4 + 5 + 6 + 6 + 8 + 9 + 10 + 11 + 13 + 14 + 16 + 17 + 19 + 21 + 23 + 24 = 220.

%p seq(floor((n+1)*(2*n^2+n+95)/144),n=0..50)

%o (Magma) [Floor((n+1)*(2*n^2+n+95)/144): n in [0..50]]; // _Vincenzo Librandi_, Apr 29 2011

%o (PARI) a(n)=(n+1)*(2*n^2+n+95)\144 \\ _Charles R Greathouse IV_, Jul 06 2017

%K nonn,easy

%O 0,3

%A _Mircea Merca_, Dec 05 2010