OFFSET
0,3
COMMENTS
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
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,-3,1,0,0,0,0,0,0,0,0,0,0,1,-3,3,-1).
FORMULA
a(n) = round((2*n+1)*(2*n^2 + 2*n + 45)/168).
a(n) = floor((2*n^3 + 3*n^2 + 46*n + 60)/84).
a(n) = ceiling((2*n^3 + 3*n^2 + 46*n - 15)/84).
a(n) = a(n-14) + (n+1)*(n-14) + 80, n > 13.
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
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
EXAMPLE
a(14) = 0 + 1 + 1 + 1 + 2 + 2 + 3 + 4 + 5 + 6 + 8 + 9 + 11 + 13 + 14 = 80.
MAPLE
seq(ceil((2*n^3+3*n^2+46*n-15)/84), n=0..50)
MATHEMATICA
Accumulate[Ceiling[Range[0, 50]^2/14]] (* Harvey P. Dale, May 14 2017 *)
PROG
(Magma) [Round((2*n+1)*(2*n^2+2*n+45)/168): n in [0..60]]; // Vincenzo Librandi, Jun 22 2011
(PARI) a(n)=(2*n^3+3*n^2+46*n+60)\84 \\ Charles R Greathouse IV, Jul 06 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Mircea Merca, Dec 05 2010
STATUS
approved