A175827 - OEIS

%I #30 Apr 02 2026 15:37:11

%S 0,1,2,3,5,8,12,17,24,33,43,56,71,88,108,131,157,186,219,256,296,341,

%T 390,443,501,564,632,705,784,869,959,1056,1159,1268,1384,1507,1637,

%U 1774,1919,2072,2232,2401,2578,2763,2957,3160,3372,3593,3824,4065,4315,4576,4847,5128,5420

%N Partial sums of ceiling(n^2/10).

%C Partial sums of A036408.

%H Vincenzo Librandi, <a href="/A175827/b175827.txt">Table of n, a(n) for n = 0..10000</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_13">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,0,0,0,0,0,0,1,-3,3,-1).

%F a(n) = round((2*n+1)*(2*n^2 + 2*n + 27)/120).

%F a(n) = floor((2*n^3 + 3*n^2 + 28*n + 36)/60).

%F a(n) = ceiling((2*n^3 + 3*n^2 + 28*n - 9)/60).

%F a(n) = a(n-10) + (n+1)*(n-10) + 43.

%F From _R. J. Mathar_, Dec 06 2010: (Start)

%F G.f.: x*(1 - x + x^3 + x^7 - x^9 + x^10) / ( (1+x)*(x^4 + x^3 + x^2 + x + 1)*(x^4 - x^3 + x^2 - x + 1)*(x-1)^4 ).

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-10) - 3*a(n-11) + 3*a(n-12) - a(n-13). (End)

%e a(10) = 0 + 1 + 1 + 1 + 2 + 3 + 4 + 5 + 7 + 9 + 10 = 43.

%p seq(ceil((2*n^3+3*n^2+28*n-9)/60),n=0..50);

%t A175827[n_] := Quotient[n*(n*(2*n + 3) + 28) + 36, 60]; Array[A175827, 60, 0] (* or *)

%t Accumulate[Ceiling[Range[0, 59]^2/10]] (* _Paolo Xausa_, Apr 01 2026 *)

%o (Magma) [Round((2*n+1)*(2*n^2+2*n+27)/120): n in [0..60]]; // _Vincenzo Librandi_, Jun 22 2011

%Y Cf. A175822, A175826.

%K nonn,easy

%O 0,3

%A _Mircea Merca_, Dec 05 2010