A177337 Partial sums of round(n^2/36).

0, 0, 0, 0, 0, 1, 2, 3, 5, 7, 10, 13, 17, 22, 27, 33, 40, 48, 57, 67, 78, 90, 103, 118, 134, 151, 170, 190, 212, 235, 260, 287, 315, 345, 377, 411, 447, 485, 525, 567, 611, 658, 707, 758, 812, 868, 927, 988, 1052, 1119, 1188

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).

Examples

			a(19) = 0 + 0 + 0 + 0 + 0 + 1 + 1 + 1 + 2 + 2 + 3 + 3 + 4 + 5 + 5 + 6 + 7 + 8 + 9 + 10 = 67.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..910
• 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+5)*(2*n^2-7*n+17)/216): n in [0..50]]; // Vincenzo Librandi, Apr 29 2011

Maple

seq(round((2*n^3+3*n^2-18*n)/216),n=0..50)

Mathematica

Accumulate[Floor[Range[0,60]^2/36+1/2]] (* Harvey P. Dale, Sep 29 2011 *)

Formula

a(n) = round((2*n+1)*(2*n^2 + 2*n - 19)/432).

a(n) = floor((n+5)*(2*n^2 - 7*n + 17)/216).

a(n) = ceiling((n-4)*(2*n^2 + 11*n + 26)/216).

a(n) = round((2*n^3 + 3*n^2 - 18*n)/216).

a(n) = a(n-36) + (n+1)*(n-36) + 447, n > 35.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-18) - 3*a(n-19) + 3*a(n-20) - a(n-21) with g.f. x^5*(1 - x + x^3 - x^4 + x^5 - x^6 + x^7 - x^9 + x^10) / ( (1+x) *(1 + x + x^2) *(x^2 - x + 1) *(x^6 + x^3 + 1) *(x^6 - x^3 + 1) *(x-1)^4 ). - R. J. Mathar, Dec 13 2010