OFFSET
0,5
COMMENTS
The round 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..885
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 + 3)/192).
a(n) = floor((n+3)*(2*n^2 - 3*n + 13)/96).
a(n) = ceiling((n-2)*(2*n^2 + 7*n + 18)/96).
a(n) = round((2*n^3 + 3*n^2 + 4*n)/96).
a(n) = a(n-16) + (n+1)*(n-16) + 94, n > 15.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-8) - 3*a(n-9) + 3*a(n-10) - a(n-11) with g.f. x^3*(1 - x + x^2 + x^4 - x^3) / ( (1+x)*(1+x^2)*(1+x^4)*(x-1)^4 ). - R. J. Mathar, Dec 13 2010
EXAMPLE
a(16) = 0 + 0 + 0 + 1 + 1 + 2 + 2 + 3 + 4 + 5 + 6 + 8 + 9 + 11 + 12 + 14 + 16 = 94.
MAPLE
seq(round((2*n^3+3*n^2+4*n)/96), n=0..50)
MATHEMATICA
Accumulate[Round[Range[0, 50]^2/16]] (* Harvey P. Dale, Mar 16 2011 *)
PROG
(Magma) [Floor((n+3)*(2*n^2-3*n+13)/96): n in [0..50]]; // Vincenzo Librandi, Apr 29 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Mircea Merca, Dec 10 2010
STATUS
approved