A175842 - OEIS

%I #31 Sep 08 2022 08:45:52

%S 0,1,2,3,5,7,10,14,19,25,33,42,53,66,80,97,116,137,161,187,216,248,

%T 283,321,363,408,457,510,566,627,692,761,835,913,996,1084,1177,1275,

%U 1379,1488,1603,1724,1850,1983,2122,2267,2419,2577,2742,2914,3093

%N Partial sums of ceiling(n^2/14).

%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="/A175842/b175842.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_17">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,0,0,0,0,0,0,0,0,0,0,1,-3,3,-1).

%F a(n) = round((2*n+1)*(2*n^2 + 2*n + 45)/168).

%F a(n) = floor((2*n^3 + 3*n^2 + 46*n + 60)/84).

%F a(n) = ceiling((2*n^3 + 3*n^2 + 46*n - 15)/84).

%F a(n) = a(n-14) + (n+1)*(n-14) + 80, n > 13.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-14) - 3*a(n-15) + 3*a(n-16) - a(n-17). - _R. J. Mathar_, Mar 11 2012

%F G.f.: x*(x^14 - x^13 + x^11 - x^10 + x^9 + x^5 - x^4 + x^3 - x + 1)/((x-1)^4*(x+1)*(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)*(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)). - _Colin Barker_, Oct 26 2012

%e a(14) = 0 + 1 + 1 + 1 + 2 + 2 + 3 + 4 + 5 + 6 + 8 + 9 + 11 + 13 + 14 = 80.

%p seq(ceil((2*n^3+3*n^2+46*n-15)/84),n=0..50)

%t Accumulate[Ceiling[Range[0,50]^2/14]] (* _Harvey P. Dale_, May 14 2017 *)

%o (Magma) [Round((2*n+1)*(2*n^2+2*n+45)/168): n in [0..60]]; // _Vincenzo Librandi_, Jun 22 2011

%o (PARI) a(n)=(2*n^3+3*n^2+46*n+60)\84 \\ _Charles R Greathouse IV_, Jul 06 2017

%K nonn,easy

%O 0,3

%A _Mircea Merca_, Dec 05 2010