A177339 Partial sums of round(n^2/44).

0, 0, 0, 0, 0, 1, 2, 3, 4, 6, 8, 11, 14, 18, 22, 27, 33, 40, 47, 55, 64, 74, 85, 97, 110, 124, 139, 156, 174, 193, 213, 235, 258, 283, 309, 337, 366, 397, 430, 465, 501, 539, 579, 621, 665, 711, 759, 809, 861, 916, 973

Offset: 0

Author

Mircea Merca, Dec 10 2010

Keywords

• nonn
• easy

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).

Examples

			a(15) = 0 + 0 + 0 + 0 + 0 + 1 + 1 + 1 + 1 + 2 + 2 + 3 + 3 + 4 + 4 + 5 = 27.
		

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.

Crossrefs

Cf. A177100, A177116.

Programs

Magma

[Round((2*n+1)*(2*n^2+2*n-15)/528): n in [0..60]]; // Vincenzo Librandi, Jun 23 2011

Maple

seq(round(n*(n-2)*(2*n+7)/264),n=0..50)

Formula

a(n) = round((2*n+1)*(2*n^2 + 2*n - 15)/528).

a(n) = floor((n+5)*(2*n^2 - 7*n + 21)/264).

a(n) = ceiling((n-4)*(2*n^2 + 11*n + 30)/264).

a(n) = round(n*(n-2)*(2*n+7)/264).

a(n) = a(n-44) + (n+1)*(n-44) + 665, n > 43.

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) + a(n-11) - 2*a(n-12) + 2*a(n-14) - a(n-15) with g.f. x^5*(1 - x^2 + x^4) / ( (1+x) *(x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) *(x-1)^4 ). - R. J. Mathar, Dec 13 2010