A123919 -id:A123919 - cp's OEIS frontend

A056827 a(n) = floor(n^2/6).

0, 0, 0, 1, 2, 4, 6, 8, 10, 13, 16, 20, 24, 28, 32, 37, 42, 48, 54, 60, 66, 73, 80, 88, 96, 104, 112, 121, 130, 140, 150, 160, 170, 181, 192, 204, 216, 228, 240, 253, 266, 280, 294, 308, 322, 337, 352, 368, 384, 400, 416, 433, 450, 468, 486, 504

Offset: 0

N. J. A. Sloane, Sep 02 2000

a(n-1) represents the floor of the area under the polygon connecting the lattice points (n, floor(n/3)) from 0..n, n>0 (see example). - Wesley Ivan Hurt, Jun 06 2014

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  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17   .. n
  0  0  0  1  2  4  6  8 10 13 16 20 24 28 32 37 42 48   .. a(n)
     0  0  0  1  2  4  6  8 10 13 16 20 24 28 32 37 42   .. a(n-1) <--

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,1,-2,1).

Cf. A000290, A007590, A000212, A002620, A118015, A118013, A056834, A130519, A056838, A056865.

List([0..60], n-> Int(n^2/6) ); # G. C. Greubel, Jul 23 2019

[Floor(n^2/6): n in [0..60]]; // Vincenzo Librandi, May 08 2011

A056827:=n->floor(n^2/6); seq(A056827(k), k=0..60); # Wesley Ivan Hurt, Oct 29 2013

Floor[Range[0,60]^2/6] (* or *) LinearRecurrence[{2,-1,0,0,0,1,-2,1}, {0,0,0,1,2,4,6,8}, 60] (* Harvey P. Dale, Jun 06 2013 *)

n^2\6 \\ Charles R Greathouse IV, May 08 2011

[floor(n^2/6) for n in (0..60)] # G. C. Greubel, Jul 23 2019

From R. J. Mathar, Nov 22 2008: (Start)
G.f.: x^3*(1+x^2)/((1+x)*(1-x)^3*(1+x+x^2)*(1-x+x^2)).
a(n+1) - a(n) = A123919(n). (End)
a(n) = floor( (1/2) * Sum_{i=1..n+1} (ceiling(i/3) + floor(i/3) - 1) ). - Wesley Ivan Hurt, Jun 06 2014
Sum_{n>=3} 1/a(n) = 15/8 + Pi^2/36 - Pi/(4*sqrt(3)) + tan(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)). - Amiram Eldar, Aug 13 2022

A123920 Number of numbers congruent to 2 or 4 mod 6 between n and 2n inclusive.

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 3, 4, 3, 4, 4, 4, 5, 6, 5, 6, 6, 6, 7, 8, 7, 8, 8, 8, 9, 10, 9, 10, 10, 10, 11, 12, 11, 12, 12, 12, 13, 14, 13, 14, 14, 14, 15, 16, 15, 16, 16, 16, 17, 18, 17, 18, 18, 18, 19, 20, 19, 20, 20, 20, 21, 22, 21, 22, 22, 22, 23, 24, 23, 24, 24, 24, 25, 26, 25, 26, 26, 26

Offset: 1

Giovanni Teofilatto, Oct 29 2006

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Cf. A123919.

a:=[1,2,1,2,2,2,3];; for n in [8..80] do a[n]:=a[n-1]+a[n-6]-a[n-7]; od; a; # G. C. Greubel, Aug 07 2019

R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( x*(1+x-x^2+x^3)/((1-x)*(1-x^6)) )); // G. C. Greubel, Aug 07 2019

seq(coeff(series(x*(1+x-x^2+x^3)/((1-x)*(1-x^6)), x, n+1), x, n), n = 1..80); # G. C. Greubel, Aug 07 2019

f[n_]:= Floor[n/2] - Floor[n/6]; Table[f[2n] - f[n-1], {n, 80}] (* Robert G. Wilson v *)
Table[Count[Range[n,2n],?(MemberQ[{2,4},Mod[#,6]]&)],{n,80}] (* _Harvey P. Dale, Mar 25 2019 *)
LinearRecurrence[{1,0,0,0,0,1,-1}, {1,2,1,2,2,2,3}, 80] (* G. C. Greubel, Aug 07 2019 *)

my(x='x+O('x^80)); Vec(x*(1+x-x^2+x^3)/((1-x)*(1-x^6))) \\ G. C. Greubel, Aug 07 2019

def A123920_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+x-x^2+x^3)/((1-x)*(1-x^6)) ).list()
a=A123920_list(80); a[1:] # G. C. Greubel, Aug 07 2019

a(n) = 2k - 1 for n = {6k - 5, 6k - 3}, where k = 1,2,3,... a(n) = 2k for n = {6k - 4, 6k - 2, 6k - 1, 6k}, where k = 1,2,3,... - Alexander Adamchuk, Nov 08 2006

G.f.: x*(1+x-x^2+x^3)/((1-x)*(1-x^6)). - G. C. Greubel, Aug 07 2019

Corrected and extended by Robert G. Wilson v, Oct 29 2006

More terms from Alexander Adamchuk, Nov 08 2006

cp's OEIS Frontend

A056827 a(n) = floor(n^2/6).

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