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A008810 a(n) = ceiling(n^2/3). 22

%I

%S 0,1,2,3,6,9,12,17,22,27,34,41,48,57,66,75,86,97,108,121,134,147,162,

%T 177,192,209,226,243,262,281,300,321,342,363,386,409,432,457,482,507,

%U 534,561,588,617,646,675,706,737,768,801,834,867,902,937,972,1009,1046

%N a(n) = ceiling(n^2/3).

%C 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

%C a(A008585(n)) = A033428(n). - _Reinhard Zumkeller_, Dec 20 2012

%C 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

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

%H Reinhard Zumkeller, <a href="/A008810/b008810.txt">Table of n, a(n) for n = 0..10000</a>

%H S. Lafortune, A. Ramani, B. Grammaticos, Y. Ohta and K.M. Tamizhmani, <a href="http://arXiv.org/abs/nlin.SI/0104020">Blending two discrete integrability criteria: ...</a>, arXiv:nlin/0104020 [nlin.SI], 2001.

%H Kival Ngaokrajang, <a href="/A008810/a008810.pdf">Illustration of initial terms</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).

%F Euler transform of length 6 sequence [ 2, 0, 2, 0, 0, -1]. - _Michael Somos_, Aug 03 2006

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

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

%F 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

%F 9*a(n) = 4 +3*n^2 -2*A099837(n+3). - _R. J. Mathar_, May 02 2013

%F a(n) = n^2 - 2*A000212(n). - _Wesley Ivan Hurt_, Jul 07 2013

%p seq(ceil(n^2/3), n=0..60); # _G. C. Greubel_, Sep 12 2019

%t Ceiling[Range[0,60]^2/3] (* _Vladimir Joseph Stephan Orlovsky_, Mar 15 2011 *)

%t LinearRecurrence[{2,-1,1,-2,1},{0,1,2,3,6},60] (* _Harvey P. Dale_, Jun 20 2011 *)

%o (PARI) a(n)=ceil(n^2/3) /* _Michael Somos_, Aug 03 2006 */

%o (Haskell)

%o a008810 = ceiling . (/ 3) . fromInteger . a000290

%o a008810_list = [0,1,2,3,6] ++ zipWith5

%o (\u v w x y -> 2 * u - v + w - 2 * x + y)

%o (drop 4 a008810_list) (drop 3 a008810_list) (drop 2 a008810_list)

%o (tail a008810_list) a008810_list

%o -- _Reinhard Zumkeller_, Dec 20 2012

%o (MAGMA) [Ceiling(n^2/3): n in [0..60]]; // _G. C. Greubel_, Sep 12 2019

%o (Sage) [ceil(n^2/3) for n in (0..60)] # _G. C. Greubel_, Sep 12 2019

%Y A056105(n)=a(3n-1). A056109(n)=a(3n+1). - _Michael Somos_, Aug 03 2006

%Y Cf. A000290.

%Y 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).

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_

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Last modified December 6 21:48 EST 2019. Contains 329809 sequences. (Running on oeis4.)