OFFSET
0,6
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..895
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 (2,0,-2,1,1,-2,0,2,-1).
FORMULA
a(n) = A001304(n-4).
a(n) = round((2*n+1)*(2*n^2 + 2*n - 15)/240).
a(n) = floor((n+4)*(2*n^2 - 5*n + 6)/120).
a(n) = ceiling((n-3)*(2*n^2 + 9*n + 13)120).
a(n) = round(n*(n-2)*(2*n+7)/120).
a(n) = a(n-20) + (n+1)*(n-20) + 141, n > 19.
From R. J. Mathar, Dec 12 2010: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) + a(n-5) - 2*a(n-6) + 2*a(n-8) - a(n-9).
G.f.: x^4 / ( (1+x)*(1+x+x^2+x^3+x^4)*(1-x)^4 ). (End)
EXAMPLE
a(20) = 0 + 0 + 0 + 0 + 1 + 1 + 2 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 10 + 11 + 13 + 14 + 16 + 18 + 20 = 141.
MAPLE
seq(round(n*(n-2)*(2*n+7)/120), n=0..50)
MATHEMATICA
f[n_] := Round[n^2/20]; Accumulate@ Array[f, 51, 0] (* Robert G. Wilson v, Dec 20 2010 *)
PROG
(Magma) [Floor((n+4)*(2*n^2-5*n+6)/120): n in [0..50]]; // Vincenzo Librandi, Apr 29 2011
(SageMath)
[(n+4)*(2*n^2 -5*n +6)//120 for n in range(56)] # G. C. Greubel, Apr 27 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Mircea Merca, Dec 10 2010
STATUS
approved