A175776 - OEIS

%I #34 May 06 2024 06:51:16

%S 0,0,0,0,1,2,4,7,11,16,22,30,39,50,63,78,95,114,135,159,185,214,246,

%T 281,319,360,405,453,505,561,621,685,753,825,902,983,1069,1160,1256,

%U 1357,1463,1575,1692,1815,1944,2079,2220,2367,2520,2680,2846

%N Partial sums of floor(n^2/15).

%C There are several sequences of integers of the form floor(n^2/k) for whose partial sums we can establish identities as following (only for k = 2,...,9,11,12,15,16,24).

%H Vincenzo Librandi, <a href="/A175776/b175776.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_18">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,0,0,0,0,0,0,0,0,0,0,0,1,-3,3,-1).

%F a(n) = round((2*n+1)*(n^2 + n - 14)/90).

%F a(n) = floor((n-2)*(2*n^2 + 7*n - 13)/90).

%F a(n) = ceiling((n+3)*(2*n^2 - 3*n - 18)/90).

%F a(n) = a(n-15) + (n+1)*(n-15) + 78, n > 14.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-15) - 3*a(n-16) + 3*a(n-17) - a(n-18); a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=1, a(5)=2, a(6)=4, a(7)=7, a(8)=11, a(9)=16, a(10)=22, a(11)=30, a(12)=39, a(13)=50, a(14)=63, a(15)=78, a(16)=95, a(17)=114. - _Harvey P. Dale_, May 04 2012

%F G.f.: x^4*(x+1)*(x^2 - x + 1)*(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)/((x-1)^4*(x^2 + x + 1)*(x^4 + x^3 + x^2 + x + 1)*(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1)). [_Colin Barker_, Oct 27 2012]

%e a(15) = 0 + 0 + 0 + 0 + 1 + 1 + 2 + 3 + 4 + 5 + 6 + 8 + 9 + 11 + 13 + 15 = 78.

%p seq(round((2*n+1)*(n^2+n-14)/90),n=0..50)

%t Accumulate[Floor[Range[0,60]^2/15]] (* or *) LinearRecurrence[{3, -3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -3, 3, -1}, {0, 0, 0, 0, 1, 2, 4, 7, 11, 16, 22, 30, 39, 50, 63, 78, 95, 114}, 60] (* _Harvey P. Dale_, May 04 2012 *)

%o (Magma) [Round((2*n+1)*(n^2+n-14)/90): n in [0..60]]; // _Vincenzo Librandi_, Jun 22 2011

%K nonn,easy

%O 0,6

%A _Mircea Merca_, Dec 04 2010