A177176 Partial sums of round(n^2/13).

0, 0, 0, 1, 2, 4, 7, 11, 16, 22, 30, 39, 50, 63, 78, 95, 115, 137, 162, 190, 221, 255, 292, 333, 377, 425, 477, 533, 593, 658, 727, 801, 880, 964, 1053, 1147, 1247, 1352, 1463, 1580, 1703, 1832, 1968, 2110, 2259, 2415, 2578, 2748, 2925, 3110, 3302

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(13) = 0 + 0 + 0 + 1 + 1 + 2 + 3 + 4 + 5 + 6 + 8 + 9 + 11 + 13 = 63.
		

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, A181120.

Programs

Magma

[Round(n*(n+1)*(2*n+1)/78): n in [0..60]]; // Vincenzo Librandi, Jun 23 2011

Maple

seq(round(n*(n+1)*(2*n+1)/78),n=0..50)

PARI

s=0;vector(90,n,s+=n^2\13)

Formula

a(n) = round(n*(n+1)*(2*n+1)/78).

a(n) = floor((n+3)*(2*n^2 - 3*n + 10)/78).

a(n) = ceiling((n-2)*(2*n^2 + 7*n + 15)/78).

a(n) = a(n-13) + (n+1)*(n-13) + 63, n > 12.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-13) - 3*a(n-14) + 3*a(n-15) - a(n-16) with g.f. x^3*(1+x)*(x^2 - x + 1)*(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1) / ( (x^12 + x^11 + 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