OFFSET
0,5
COMMENTS
The round function, also called the nearest integer function, is defined here by round(x)=floor(x+1/2).
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).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..870
Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,0,0,0,0,0,1,-3,3,-1).
FORMULA
a(n) = round((n-2)*(n+3)*(2*n+1)/66).
a(n) = floor((2*n^3 + 3*n^2 - 11*n + 18)/66).
a(n) = ceiling((2*n^3 + 3*n^2 - 11*n - 30)/66).
a(n) = round(n*(2*n^2 + 3*n - 11)/66).
a(n) = a(n-11) + (n+1)*(n-11) + 44, n > 10.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-11) - 3*a(n-12) + 3*a(n-13) - a(n-14). - R. J. Mathar, Dec 10 2010
G.f.: x^3 *(1+x) *(x^2-x+1) *(x^4-x^3+x^2-x+1) / ( (x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) *(x-1)^4 ). - R. J. Mathar, Dec 10 2010 [Typo fixed by Colin Barker, Oct 10 2012]
EXAMPLE
a(11) = 0 + 0 + 0 + 1 + 1 + 2 + 3 + 4 + 6 + 7 + 9 + 11 = 44.
MAPLE
seq(round((2*n^3+3*n^2-11*n)/66), n=0..50)
MATHEMATICA
Accumulate[Round[Range[0, 50]^2/11]] (* or *) LinearRecurrence[{3, -3, 1, 0, 0, 0, 0, 0, 0, 0, 1, -3, 3, -1}, {0, 0, 0, 1, 2, 4, 7, 11, 17, 24, 33, 44, 57, 72}, 60] (* Harvey P. Dale, Dec 10 2014 *)
PROG
(PARI) a(n)=(2*n^3+3*n^2-11*n+18)\66 \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Mircea Merca, Dec 09 2010
STATUS
approved