OFFSET
0,3
COMMENTS
Partial sums of A008747.
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).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,0,2,-1).
Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
FORMULA
a(n) = round((2*n+1)*(2*n^2 + 2*n + 17)/72).
a(n) = floor((n+1)*(2*n^2 + n + 17)/36).
a(n) = ceiling((2*n^3 + 3*n^2 + 18*n)/36).
a(n) = round((2*n^3 + 3*n^2 + 18*n)/36).
a(n) = a(n-6) + (n+1)*(n-6) + 18, n > 5.
From Mircea Merca, Jan 09 2011: (Start)
a(n) = 2*a(n-1) - a(n-3) - a(n-4) + 2*a(n-6) - a(n-7), n > 6.
G.f.: x*(x^4+1) / ( (x+1)*(x^2+x+1)*(x-1)^4 ). (End)
EXAMPLE
a(6) = 0 + 1 + 1 + 2 + 3 + 5 + 6 = 18.
MAPLE
seq(floor((n+1)*(2*n^2+n+17)/36), n=0..50)
MATHEMATICA
Accumulate[Ceiling[Range[0, 50]^2/6]] (* Harvey P. Dale, Jan 17 2016 *)
PROG
(Magma) [Round((2*n+1)*(2*n^2+2*n+17)/72): n in [0..60]]; // Vincenzo Librandi, Jun 22 2011
(PARI) a(n) = (n+1)*(2*n^2+n+17)\36; \\ Altug Alkan, Sep 21 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Mircea Merca, Dec 05 2010
STATUS
approved