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Partial sums of floor(n/6).
15

%I #49 Aug 09 2024 11:21:53

%S 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,

%T 52,56,60,65,70,75,80,85,90,96,102,108,114,120,126,133,140,147,154,

%U 161,168,176,184,192

%N Partial sums of floor(n/6).

%C Partial sums of A152467.

%H Vincenzo Librandi, <a href="/A174709/b174709.txt">Table of n, a(n) for n = 0..10000</a>

%H Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Merca/merca3.html">Inequalities and Identities Involving Sums of Integer Functions</a>, J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,1,-2,1).

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

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

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

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

%F From _R. J. Mathar_, Nov 30 2010: (Start)

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

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

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

%F 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

%F a(n) = A008724(n-2). - _R. J. Mathar_, Jul 10 2015

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

%e 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.

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

%t Table[Sum[Floor[k/6], {k,0,n}], {n,0,25}] (* _G. C. Greubel_, Dec 13 2016 *)

%o (Magma) [Round(n*(n-4)/12): n in [0..60]]; // _Vincenzo Librandi_, Jun 22 2011

%o (PARI) a(n)=(n-2)^2\12 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A008724, A152467.

%K nonn,easy

%O 0,8

%A _Mircea Merca_, Nov 30 2010