A174709 Partial sums of floor(n/6).

0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 18, 21, 24, 27, 30, 33, 36, 40, 44, 48, 52, 56, 60, 65, 70, 75, 80, 85, 90, 96, 102, 108, 114, 120, 126, 133, 140, 147, 154, 161, 168, 176, 184, 192

Offset: 0

Author

Mircea Merca, Nov 30 2010

Keywords

• nonn
• easy

Comments

Partial sums of A152467.

Examples

			a(7) = floor(0/6) + floor(1/6) + floor(2/6) + floor(3/6) + floor(4/6) + floor(5/6) + floor(6/6) + floor(7/6) = 0 + 0 + 0 + 0 + 0 + 0 + 1 + 1 = 2.
		

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.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,1,-2,1).

Crossrefs

Cf. A008724, A152467.

Programs

Magma

[Round(n*(n-4)/12): n in [0..60]]; // Vincenzo Librandi, Jun 22 2011

Maple

a(n):=round(n*(n-4)/12)

Mathematica

Table[Sum[Floor[k/6], {k,0,n}], {n,0,25}] (* G. C. Greubel, Dec 13 2016 *)

PARI

a(n)=(n-2)^2\12 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = round(n*(n-4)/12) = round((2*n^2 - 8*n - 1)/24).

a(n) = floor((n-2)^2/12).

a(n) = ceiling((n+1)*(n-5)/12).

a(n) = a(n-6) + n - 5, n > 5.

From R. J. Mathar, Nov 30 2010: (Start)

a(n) = 2*a(n-1) - a(n-2) + a(n-6) - 2*a(n-7) + a(n-8).

G.f.: -x^6 / ( (1+x)*(x^2-x+1)*(1+x+x^2)*(x-1)^3 ).

a(n) = -n/3 + 5/72 + n^2/12 + (-1)^n/24 + A057079(n+5)/6 + A061347(n)/18. (End)

a(6n) = A000567(n), a(6n+1) = 2*A000326(n), a(6n+2) = A033428(n), a(6n+3) = A049451(n), a(6n+4) = A045944(n), a(6n+5) = A028896(n). - Philippe Deléham, Mar 26 2013

a(n) = A008724(n-2). - R. J. Mathar, Jul 10 2015

Sum_{n>=6} 1/a(n) = Pi^2/18 - Pi/(2*sqrt(3)) + 49/12. - Amiram Eldar, Aug 13 2022