OFFSET
0,5
COMMENTS
Quasipolynomial.
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,-3,1,0,0,1,-3,3,-1).
FORMULA
a(n) = Sum_{k=0..n} floor(k^2/6).
a(n) = round((2*n^3 + 3*n^2 - 12*n - 6)/36).
a(n) = round((4*n^3 + 6*n^2 - 24*n - 13)/72).
a(n) = floor((2*n^3 + 3*n^2 - 12*n + 7)/36).
a(n) = ceiling((2*n^3 + 3*n^2 - 12*n - 20)/36).
a(n) = a(n-6) + n^2 - 5*n + 7, n > 5.
G.f.: x^3*(1+x^2) / ( (1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^4 ). - R. J. Mathar, Jan 28 2012
EXAMPLE
a(5) = 7 = 0 + 0 + 0 + 1 + 2 + 4.
MAPLE
a(n):=round((2*n^3 +3*n^2 -12*n-6)/36)
MATHEMATICA
Accumulate[Floor[Range[0, 50]^2/6]] (* Harvey P. Dale, Jul 30 2020 *)
PROG
(Magma) [Round((2*n^3+3*n^2-12*n-6)/36): n in [0..60]]; // Vincenzo Librandi, Jun 25 2011
(PARI) a(n)=(2*n^3+3*n^2-12*n+7)\36 \\ Charles R Greathouse IV, Jan 29 2012
(Python)
a171965 = [0]
for n in range(1, 60): a171965.append(a171965[-1] + n*n//6)
print(a171965) # Gennady Eremin, Mar 13 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Mircea Merca, Nov 19 2010
STATUS
approved