This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A008810 #75 Oct 29 2022 09:28:14
%S A008810 0,1,2,3,6,9,12,17,22,27,34,41,48,57,66,75,86,97,108,121,134,147,162,
%T A008810 177,192,209,226,243,262,281,300,321,342,363,386,409,432,457,482,507,
%U A008810 534,561,588,617,646,675,706,737,768,801,834,867,902,937,972,1009,1046
%N A008810 a(n) = ceiling(n^2/3).
%C A008810 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
%C A008810 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
%D A008810 J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, number of red blocks in Fig 2.5.
%H A008810 Reinhard Zumkeller, <a href="/A008810/b008810.txt">Table of n, a(n) for n = 0..10000</a>
%H A008810 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 A008810 Kival Ngaokrajang, <a href="/A008810/a008810.pdf">Illustration of initial terms</a>.
%H A008810 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F A008810 a(-n) = a(n) = ceiling(n^2/3).
%F A008810 G.f.: x*(1 + x^3)/((1 - x)^2*(1 - x^3)) = x*(1 - x^6)/((1 - x)*(1 - x^3))^2.
%F A008810 From _Michael Somos_, Aug 03 2006: (Start)
%F A008810 Euler transform of length 6 sequence [ 2, 0, 2, 0, 0, -1].
%F A008810 a(3n-1) = A056105(n).
%F A008810 a(3n+1) = A056109(n). (End)
%F A008810 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
%F A008810 a(A008585(n)) = A033428(n). - _Reinhard Zumkeller_, Dec 20 2012
%F A008810 9*a(n) = 4 + 3*n^2 - 2*A099837(n+3). - _R. J. Mathar_, May 02 2013
%F A008810 a(n) = n^2 - 2*A000212(n). - _Wesley Ivan Hurt_, Jul 07 2013
%F A008810 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
%F A008810 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
%p A008810 seq(ceil(n^2/3), n=0..60); # _G. C. Greubel_, Sep 12 2019
%t A008810 Ceiling[Range[0,60]^2/3] (* _Vladimir Joseph Stephan Orlovsky_, Mar 15 2011 *)
%t A008810 LinearRecurrence[{2,-1,1,-2,1},{0,1,2,3,6},60] (* _Harvey P. Dale_, Jun 20 2011 *)
%o A008810 (PARI) a(n)=ceil(n^2/3) /* _Michael Somos_, Aug 03 2006 */
%o A008810 (Haskell)
%o A008810 a008810 = ceiling . (/ 3) . fromInteger . a000290
%o A008810 a008810_list = [0,1,2,3,6] ++ zipWith5
%o A008810 (\u v w x y -> 2 * u - v + w - 2 * x + y)
%o A008810 (drop 4 a008810_list) (drop 3 a008810_list) (drop 2 a008810_list)
%o A008810 (tail a008810_list) a008810_list
%o A008810 -- _Reinhard Zumkeller_, Dec 20 2012
%o A008810 (Magma) [Ceiling(n^2/3): n in [0..60]]; // _G. C. Greubel_, Sep 12 2019
%o A008810 (Sage) [ceil(n^2/3) for n in (0..60)] # _G. C. Greubel_, Sep 12 2019
%Y A008810 Cf. A000290, A056105, A056109.
%Y A008810 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 A008810 nonn,easy,nice
%O A008810 0,3
%A A008810 _N. J. A. Sloane_