|
|
A175812
|
|
Partial sums of ceiling(n^2/6).
|
|
1
|
|
|
0, 1, 2, 4, 7, 12, 18, 27, 38, 52, 69, 90, 114, 143, 176, 214, 257, 306, 360, 421, 488, 562, 643, 732, 828, 933, 1046, 1168, 1299, 1440, 1590, 1751, 1922, 2104, 2297, 2502, 2718, 2947, 3188, 3442, 3709, 3990, 4284, 4593, 4916, 5254, 5607, 5976, 6360, 6761, 7178
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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.
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
|
|
|
STATUS
|
approved
|
|
|
|