A008810 a(n) = ceiling(n^2/3).

0, 1, 2, 3, 6, 9, 12, 17, 22, 27, 34, 41, 48, 57, 66, 75, 86, 97, 108, 121, 134, 147, 162, 177, 192, 209, 226, 243, 262, 281, 300, 321, 342, 363, 386, 409, 432, 457, 482, 507, 534, 561, 588, 617, 646, 675, 706, 737, 768, 801, 834, 867, 902, 937, 972, 1009, 1046

Offset

0,3

Comments

a(n+1) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and 3*w = 2*x + y. - Clark Kimberling, Jun 04 2012

a(n) is also the number of L-shapes (3-cell polyominoes) packing into an n X n square. See illustration in links. - Kival Ngaokrajang, Nov 10 2013

References

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, number of red blocks in Fig 2.5.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

S. Lafortune, A. Ramani, B. Grammaticos, Y. Ohta and K.M. Tamizhmani, Blending two discrete integrability criteria: ..., arXiv:nlin/0104020 [nlin.SI], 2001.

Kival Ngaokrajang, Illustration of initial terms.

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

Formula

a(-n) = a(n) = ceiling(n^2/3).

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

From Michael Somos, Aug 03 2006: (Start)

Euler transform of length 6 sequence [ 2, 0, 2, 0, 0, -1].

a(3n-1) = A056105(n).

a(3n+1) = A056109(n). (End)

a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n > 4. - Harvey P. Dale, Jun 20 2011

a(A008585(n)) = A033428(n). - Reinhard Zumkeller, Dec 20 2012

9*a(n) = 4 + 3*n^2 - 2*A099837(n+3). - R. J. Mathar, May 02 2013

a(n) = n^2 - 2*A000212(n). - Wesley Ivan Hurt, Jul 07 2013

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

E.g.f.: (exp(x)*(4 + 3*x*(1 + x)) - 4*exp(-x/2)*cos(sqrt(3)*x/2))/9. - Stefano Spezia, Oct 28 2022

Maple

seq(ceil(n^2/3), n=0..60); # G. C. Greubel, Sep 12 2019

Mathematica

Ceiling[Range[0, 60]^2/3] (* Vladimir Joseph Stephan Orlovsky, Mar 15 2011 *)
LinearRecurrence[{2, -1, 1, -2, 1}, {0, 1, 2, 3, 6}, 60] (* Harvey P. Dale, Jun 20 2011 *)

Prog

(PARI) a(n)=ceil(n^2/3) /* Michael Somos, Aug 03 2006 */
(Haskell)
a008810 = ceiling . (/ 3) . fromInteger . a000290
a008810_list = [0, 1, 2, 3, 6] ++ zipWith5
               (\u v w x y -> 2 * u - v + w - 2 * x + y)
   (drop 4 a008810_list) (drop 3 a008810_list) (drop 2 a008810_list)
   (tail a008810_list) a008810_list
-- Reinhard Zumkeller, Dec 20 2012
(Magma) [Ceiling(n^2/3): n in [0..60]]; // G. C. Greubel, Sep 12 2019
(Sage) [ceil(n^2/3) for n in (0..60)] # G. C. Greubel, Sep 12 2019

Crossrefs

Cf. A000290, A056105, A056109.

Cf. Expansions of the form (1+x^m)/((1-x)^2*(1-x^m)): A000290 (m=1), A000982 (m=2), this sequence (m=3), A008811 (m=4), A008812 (m=5), A008813 (m=6), A008814 (m=7), A008815 (m=8), A008816 (m=9), A008817 (m=10).

Sequence in context: A140495 A174873 A213172 * A280984 A339485 A176893

Adjacent sequences: A008807 A008808 A008809 * A008811 A008812 A008813

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved