A173691 Partial sums of round(n^2/6).

0, 0, 1, 3, 6, 10, 16, 24, 35, 49, 66, 86, 110, 138, 171, 209, 252, 300, 354, 414, 481, 555, 636, 724, 820, 924, 1037, 1159, 1290, 1430, 1580, 1740, 1911, 2093, 2286, 2490, 2706, 2934, 3175, 3429, 3696, 3976, 4270, 4578, 4901, 5239, 5592, 5960, 6344, 6744, 7161

Offset: 0

Author

Mircea Merca, Nov 25 2010

Keywords

• nonn
• easy

Comments

Partial sums of A056829.

Examples

			a(5) = round(1/6) + round(4/6) + round(9/6) + round(16/6) + round(25/6) = 0 + 1 + 2 + 3 + 4.
Note that 9/6 = 1.5 is rounded up.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..5000
• 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 (3,-3,1,0,0,1,-3,3,-1).

Crossrefs

Cf. A056829.

Programs

Magma

[Floor((2*n^3+3*n^2+6*n+16)/36): n in [0..60]]; // Vincenzo Librandi, Jun 22 2011

Maple

a(n):=round((2*n^(3)+3*n^(2)+6*n)/(36))

Mathematica

Accumulate[Round[Range[0, 50]^2/6]] (* or *) LinearRecurrence[{3,-3,1,0, 0,1,-3,3,-1}, {0,0,1,3,6,10,16,24,35}, 60] (* Harvey P. Dale, Jan 08 2014 *)
CoefficientList[Series[x^2(1+x^4)/((1+x)(1-x+x^2)(1+x+x^2)(1-x)^4), {x, 0, 60}], x] (* Vincenzo Librandi, Mar 26 2014 *)

PARI

vector(60, n, n--; (16+6*n+3*n^2+2*n^3)\36) \\ G. C. Greubel, Jul 02 2019

Sage

[floor((16+6*n+3*n^2+2*n^3)/36) for n in (0..60)] # G. C. Greubel, Jul 02 2019

Formula

a(n) = Sum_{k=0..n} round(k^2/6).

a(n) = round((2*n^3 + 3*n^2 + 6*n)/36).

a(n) = round((4*n^3 + 6*n^2 + 12*n + 5)/72).

a(n) = floor((2*n^3 + 3*n^2 + 6*n + 16)/36).

a(n) = ceiling((2*n^3 + 3*n^2 + 6*n - 11)/36).

a(n) = a(n-6) + n^2 - 5*n + 10, n > 5.

G.f.: x^2*(1+x^4)/((1+x)*(1-x+x^2)*(1+x+x^2)*(1-x)^4). - Bruno Berselli, Jan 12 2011