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A008810 a(n) = ceiling(n^2/3). 22
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

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

a(n) is also the number of L-shape (3 boxes polyomino) packing into 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

Euler transform of length 6 sequence [ 2, 0, 2, 0, 0, -1]. - Michael Somos, Aug 03 2006

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.

a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5), 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

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

A056105(n)=a(3n-1). A056109(n)=a(3n+1). - Michael Somos, Aug 03 2006

Cf. A000290.

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: A174873 A213172 A280984 * A176893 A144677 A309677

Adjacent sequences:  A008807 A008808 A008809 * A008811 A008812 A008813

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 15 22:25 EDT 2019. Contains 328038 sequences. (Running on oeis4.)