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%I #136 Sep 29 2022 07:41:38
%S 0,1,1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,11,12,13,13,14,15,15,16,17,17,18,
%T 19,19,20,21,21,22,23,23,24,25,25,26,27,27,28,29,29,30,31,31,32,33,33,
%U 34,35,35,36,37,37,38,39,39,40,41,41,42,43,43,44,45,45,46,47,47
%N One even number followed by two odd numbers.
%C Maximal number of points on a triangular grid of edge length n-1 with no 2 points on same row, column, or diagonal. See Problem 252 in The Inquisitive Problem Solver. - _R. K. Guy_ [Comment revised by _N. J. A. Sloane_, Jul 01 2016]
%C See also Problem C2 of 2009 International Mathematical Olympiad. - _Ruediger Jehn_, Oct 19 2021
%C Dimension of the space of weight 2n+4 cusp forms for Gamma_0(3).
%C Starting at 3, 3, ..., gives maximal number of acute angles in an n-gon. - Takenov Nurdin (takenov_vert(AT)e-mail.ru), Mar 04 2003
%C Let b(1) = b(2) = 1, b(k) = b(k-1)+( b(k-2) reduced (mod 2)); then a(n) = b(n-1). - _Benoit Cloitre_, Aug 14 2002
%C (1+x+x^2+x^3 ) / ( (1-x^2)*(1-x^3)) is the Poincaré series [or Poincare series] (or Molien series) for Sigma_4.
%C For n > 6, maximum number of knight moves to reach any square from the corner of an (n-2) X (n-2) chessboard. Likewise for n > 6, the maximum number of knight moves to reach any square from the middle of an (2n-5) X (2n-5) chessboard. - _Ralf Stephan_, Sep 15 2004
%C A transform of the Jacobsthal numbers A001045 under the mapping of g.f.s g(x)->g(x/(1+x^2)). - _Paul Barry_, Jan 16 2005
%C For n >= 1; a(n) = number of successive terms of A040001 that add to n; or length of n-th term of A028359. - _Jaroslav Krizek_, Mar 28 2010
%C For n > 0: a(n) = length of n-th row in A082870. - _Reinhard Zumkeller_, Apr 13 2014
%C Also the independence number of the n-triangular honeycomb queen graph. - _Eric W. Weisstein_, Jul 14 2017
%D J. Kurschak, Hungarian Mathematical Olympiads, 1976, Mir, Moscow.
%D Paul Vanderlind, Richard K. Guy, and Loren C. Larson, The Inquisitive Problem Solver, MAA, 2002. See Problem 252.
%H Vincenzo Librandi, <a href="/A004396/b004396.txt">Table of n, a(n) for n = 0..10000</a>
%H A. Adem and R. J. Milgram, <a href="https://archive.org/details/cohomologyoffini0000adem/page/246/mode/2up">Cohomology of Finite Groups</a>, Springer-Verlag, 2nd. ed., 2004; p. 246.
%H Art of Problem Solving Forum, <a href="http://www.artofproblemsolving.com/Forum/viewtopic.php?t=269266">Ordered triples choosing</a> - From _Joel B. Lewis_, May 21 2009
%H J. Choi and N. Pippenger, <a href="http://arxiv.org/abs/1310.1357">Counting the Angels and Devils in Escher's Circle Limit IV</a>, arXiv preprint arXiv:1310.1357 [math.CO], 2013.
%H C. L. Mallows and N. J. A. Sloane, <a href="http://dx.doi.org/10.1016/0012-365X(74)90085-5">Weight enumerators of self-orthogonal codes</a>, Discrete Math., 9 (1974), 391-400 (see proof of Theorem 1).
%H Gabriel Nivasch and Eyal Lev, <a href="https://www.jstor.org/stable/30044202">Nonattacking Queens on a Triangle</a>, Mathematics Magazine, Vol. 78, No. 5 (Dec., 2005), pp. 399-403. See Eq. (4).
%H John A. Pelesko, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Pelesko/pel11.html">Generalizing the Conway-Hofstadter $10,000 Sequence</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.5.
%H William A. Stein, <a href="http://wstein.org/Tables/dimskg0n.gp">Dimensions of the spaces S_k(Gamma_0(N))</a>.
%H William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependenceNumber.html">Independence Number</a>.
%H 50th International Mathematical Olympiad 2009, <a href="https://www.imo-official.org/problems/IMO2009SL.pdf">Problem Shortlist with Solutions</a>.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%H <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>.
%F G.f.: (x+x^3)/((1-x)*(1-x^3)).
%F a(n) = floor( (2*n + 1)/3 ).
%F a(n) = a(n-1) + (1/2)*((-1)^floor((4*n+2)/3) + 1), a(0) = 0. - Mario Catalani (mario.catalani(AT)unito.it), Oct 20 2003
%F a(n) = 2n/3 - cos(2*Pi*n/3 + Pi/3)/3 + sqrt(3)*sin(2*Pi*n/3 + Pi/3)/9. - _Paul Barry_, Mar 18 2004
%F a(n) = A096777(n+1) - A096777(n) for n > 0. - _Reinhard Zumkeller_, Jul 09 2004
%F From _Paul Barry_, Jan 16 2005: (Start)
%F G.f.: x*(1+x^2)/(1-x-x^3+x^4).
%F a(n) = a(n-1) + a(n-3) - a(n-4) for n>3.
%F a(n) = Sum_{k = 0..n} binomial(n-k-1, k)*(-1)^k*A001045(n-2k). (End)
%F a(n) = (A006369(n) - (A006369(n) mod 2) * (-1)^(n mod 3)) / (1 + A006369(n) mod 2). - _Reinhard Zumkeller_, Jan 23 2005
%F a(n) = A004773(n) - A004523(n). - _Reinhard Zumkeller_, Aug 29 2005
%F a(n) = floor(n/3) + ceiling(n/3). - _Jonathan Vos Post_, Mar 19 2006
%F a(n+1) = A008620(2n). - _Philippe Deléham_, Dec 14 2006
%F a(A032766(n)) = n. - _Reinhard Zumkeller_, Oct 30 2009
%F a(n) = floor((2*n^2+4*n+2)/(3*n+4)). - _Gary Detlefs_, Jul 13 2010
%F Euler transform of length 4 sequence [1, 1, 1, -1]. - _Michael Somos_, Jul 03 2014
%F a(n) = n - floor((n+1)/3). - _Wesley Ivan Hurt_, Sep 17 2015
%F a(n) = A092200(n) - floor((n+5)/3). - _Filip Zaludek_, Oct 27 2016
%F a(n) = -a(-n) for all n in Z. - _Michael Somos_, Oct 30 2016
%F E.g.f.: (2/9)*(3*exp(x)*x + sqrt(3)*exp(-x/2)*sin(sqrt(sqrt(3)*x/2)). - _Stefano Spezia_, Sep 20 2022
%F Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/2. - _Amiram Eldar_, Sep 29 2022
%e G.f. = x + x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 5*x^8 + 6*x^9 + 7*x^10 + ...
%p A004396:=n->floor((2*n + 1)/3); seq(A004396(n), n=0..100); # _Wesley Ivan Hurt_, Nov 30 2013
%t Table[Floor[(2 n + 1)/3], {n, 0, 75}]
%t With[{n = 50}, Riffle[Range[0, n], Range[1, n, 2], {3, -1, 3}]] (* _Harvey P. Dale_, May 14 2015 *)
%t CoefficientList[Series[(x + x^3)/((1 - x) (1 - x^3)), {x, 0, 71}], x] (* _Michael De Vlieger_, Oct 27 2016 *)
%t a[ n_] := Quotient[2 n + 1, 3]; (* _Michael Somos_, Oct 23 2017 )
%t a[ n_] := Sign[n] SeriesCoefficient[ (x + x^3) / ((1 - x) (1 - x^3)), {x, 0, Abs@n}]; (* _Michael Somos_, Oct 23 2017 *)
%t LinearRecurrence[{1, 0, 1, -1}, {1, 1, 2, 3}, {0, 20}] (* _Eric W. Weisstein_, Jul 14 2017 *)
%o (Magma) [(Floor(n/3) + Ceiling(n/3)): n in [0..70]]; // _Vincenzo Librandi_, Aug 07 2011
%o (PARI) a(n)=2*n\/3 \\ _Charles R Greathouse IV_, Apr 17 2012
%o (Haskell)
%o a004396 n = a004396_list !! n
%o a004396_list = 0 : 1 : 1 : map (+ 2) a004396_list
%o -- _Reinhard Zumkeller_, Nov 06 2012
%o (Sage) def a(n) : return( dimension_cusp_forms( Gamma0(3), 2*n+4) ); # _Michael Somos_, Jul 03 2014
%Y Cf. A001045, A002620, A004523, A004773, A006369, A008620, A032766, A040001, A082870, A092200, A096777.
%K nonn,easy
%O 0,4
%A _N. J. A. Sloane_
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