A177277 Partial sums of round(n^2/28).

0, 0, 0, 0, 1, 2, 3, 5, 7, 10, 14, 18, 23, 29, 36, 44, 53, 63, 75, 88, 102, 118, 135, 154, 175, 197, 221, 247, 275, 305, 337, 371, 408, 447, 488, 532, 578, 627, 679, 733, 790, 850, 913, 979, 1048, 1120, 1196, 1275, 1357, 1443, 1532

Offset: 0

Author

Mircea Merca, Dec 10 2010

Keywords

• nonn
• easy

Comments

The round function, also called the nearest integer 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).

Examples

			a(28) = 0 + 0 + 0 + 1 + 1 + 1 + 2 + 2 + 3 + 4 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 12 + 13 + 14 + 16 + 17 + 19 + 21 + 22 + 24 + 26 + 28 = 275.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..905
• Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.

Crossrefs

Cf. A177100, A177116.

Programs

Magma

[Floor((n+4)*(2*n^2-5*n+18)/168): n in [0..50]]; // Vincenzo Librandi, Apr 29 2011

Maple

seq(round(n*(n+2)*(2*n-1)/168),n=0..50)

Mathematica

Accumulate[Floor[Range[0,50]^2/28+1/2]] (* Harvey P. Dale, Feb 02 2012 *)

Formula

a(n) = round((2*n+1)*(2*n^2 + 2*n - 3)/336).

a(n) = floor((n+4)*(2*n^2 - 5*n + 18)/168).

a(n) = ceiling((n-3)*(2*n^2 + 9*n + 25)/168).

a(n) = round(n*(n+2)*(2*n-1)/168).

a(n) = a(n-28) + (n+1)*(n-28) + 275, n > 27.

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) with g.f. x^4*(1 - x + x^3 - x^4 + x^5 - x^7 + x^8) / ( (1+x)*(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)*(x-1)^4 ). - R. J. Mathar, Dec 12 2010