A177239 - OEIS

%I #40 Apr 28 2024 02:07:42

%S 0,0,0,0,1,2,4,6,9,13,18,24,31,39,49,60,73,87,103,121,141,163,187,213,

%T 242,273,307,343,382,424,469,517,568,622,680,741,806,874,946,1022,

%U 1102,1186,1274,1366,1463,1564,1670,1780,1895,2015,2140

%N Partial sums of round(n^2/20).

%C The round function is defined here by round(x) = floor(x + 1/2).

%C There are several sequences of integers of the form round(n^2/k) for whose partial sums we can establish identities as following (only for k = 2, ..., 9, 11, 12, 13, 16, 17, 19, 20, 28, 29, 36, 44).

%H Vincenzo Librandi, <a href="/A177239/b177239.txt">Table of n, a(n) for n = 0..895</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 (2,0,-2,1,1,-2,0,2,-1).

%F a(n) = A001304(n-4).

%F a(n) = round((2*n+1)*(2*n^2 + 2*n - 15)/240).

%F a(n) = floor((n+4)*(2*n^2 - 5*n + 6)/120).

%F a(n) = ceiling((n-3)*(2*n^2 + 9*n + 13)120).

%F a(n) = round(n*(n-2)*(2*n+7)/120).

%F a(n) = a(n-20) + (n+1)*(n-20) + 141, n > 19.

%F From _R. J. Mathar_, Dec 12 2010: (Start)

%F a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) + a(n-5) - 2*a(n-6) + 2*a(n-8) - a(n-9).

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

%e a(20) = 0 + 0 + 0 + 0 + 1 + 1 + 2 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 10 + 11 + 13 + 14 + 16 + 18 + 20 = 141.

%p seq(round(n*(n-2)*(2*n+7)/120),n=0..50)

%t f[n_] := Round[n^2/20]; Accumulate@ Array[f, 51, 0] (* _Robert G. Wilson v_, Dec 20 2010 *)

%o (Magma) [Floor((n+4)*(2*n^2-5*n+6)/120): n in [0..50]]; // _Vincenzo Librandi_, Apr 29 2011

%o (SageMath)

%o [(n+4)*(2*n^2 -5*n +6)//120 for n in range(56)] # _G. C. Greubel_, Apr 27 2024

%Y Cf. A001304, A177100, A177116.

%K nonn,easy

%O 0,6

%A _Mircea Merca_, Dec 10 2010