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
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
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved