A004396 One even number followed by two odd numbers.

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, 19, 19, 20, 21, 21, 22, 23, 23, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 37, 37, 38, 39, 39, 40, 41, 41, 42, 43, 43, 44, 45, 45, 46, 47, 47

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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]

See also Problem C2 of 2009 International Mathematical Olympiad. - Ruediger Jehn, Oct 19 2021

Dimension of the space of weight 2n+4 cusp forms for Gamma_0(3).

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

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

(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.

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

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

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

For n > 0: a(n) = length of n-th row in A082870. - Reinhard Zumkeller, Apr 13 2014

Also the independence number of the n-triangular honeycomb queen graph. - Eric W. Weisstein, Jul 14 2017

In a game of basketball points can be accumulated by making field goals (two or three points) or free throws (one point). a(n) is the number of different ways to score n-1 points. For example, a score of 4 can be achieved in 3 different ways, with 2 shots (3+1 or 2+2), 3 shots (2+1+1) or 4 shots (1+1+1+1), so a(5) = 3. - Ivan N. Ianakiev, Mar 31 2025

Examples

			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 + ...
		

References

• J. Kurschak, Hungarian Mathematical Olympiads, 1976, Mir, Moscow.
• Paul Vanderlind, Richard K. Guy, and Loren C. Larson, The Inquisitive Problem Solver, MAA, 2002. See Problem 252.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 246.
• Art of Problem Solving Forum, Ordered triples choosing. - _Joel B. Lewis_, May 21 2009
• J. Choi and N. Pippenger, Counting the Angels and Devils in Escher's Circle Limit IV, arXiv preprint arXiv:1310.1357 [math.CO], 2013.
• C. L. Mallows and N. J. A. Sloane, Weight enumerators of self-orthogonal codes, Discrete Math., 9 (1974), 391-400 (see proof of Theorem 1).
• Gabriel Nivasch and Eyal Lev, Nonattacking Queens on a Triangle, Mathematics Magazine, Vol. 78, No. 5 (Dec., 2005), pp. 399-403. See Eq. (4).
• John A. Pelesko, Generalizing the Conway-Hofstadter $10,000 Sequence, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.5.
• William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
• William A. Stein, The modular forms database
• Eric Weisstein's World of Mathematics, Independence Number.
• 50th International Mathematical Olympiad 2009, Problem Shortlist with Solutions.
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
• Index to sequences related to Olympiads.

Crossrefs

Cf. A001045, A002620, A004523, A004773, A006369, A008620, A032766, A040001, A082870, A092200, A096777.

Programs

Haskell

a004396 n = a004396_list !! n
a004396_list = 0 : 1 : 1 : map (+ 2) a004396_list
-- Reinhard Zumkeller, Nov 06 2012

Magma

[(Floor(n/3) + Ceiling(n/3)): n in [0..70]]; // Vincenzo Librandi, Aug 07 2011

Maple

A004396:=n->floor((2*n + 1)/3); seq(A004396(n), n=0..100); # Wesley Ivan Hurt, Nov 30 2013

Mathematica

Table[Floor[(2 n + 1)/3], {n, 0, 75}]
With[{n = 50}, Riffle[Range[0, n], Range[1, n, 2], {3, -1, 3}]] (* Harvey P. Dale, May 14 2015 *)
CoefficientList[Series[(x + x^3)/((1 - x) (1 - x^3)), {x, 0, 71}], x] (* Michael De Vlieger, Oct 27 2016 *)
a[ n_] := Quotient[2 n + 1, 3]; (* Michael Somos, Oct 23 2017 *)
a[ n_] := Sign[n] SeriesCoefficient[ (x + x^3) / ((1 - x) (1 - x^3)), {x, 0, Abs@n}]; (* Michael Somos, Oct 23 2017 *)
LinearRecurrence[{1, 0, 1, -1}, {1, 1, 2, 3}, {0, 20}] (* Eric W. Weisstein, Jul 14 2017 *)
f[-1]=0; f[n_]:=Length[Union[Plus@@@FrobeniusSolve[{1,2,3},n]]]; f/@Range[-1,100] (* Ivan N. Ianakiev, Mar 31 2025 *)

PARI

a(n)=2*n\/3 \\ Charles R Greathouse IV, Apr 17 2012

Sage

def a(n) : return( dimension_cusp_forms( Gamma0(3), 2*n+4) ); # Michael Somos, Jul 03 2014

Formula

G.f.: (x+x^3)/((1-x)*(1-x^3)).

a(n) = floor( (2*n + 1)/3 ).

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

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

a(n) = A096777(n+1) - A096777(n) for n > 0. - Reinhard Zumkeller, Jul 09 2004

From Paul Barry, Jan 16 2005: (Start)

G.f.: x*(1+x^2)/(1-x-x^3+x^4).

a(n) = a(n-1) + a(n-3) - a(n-4) for n>3.

a(n) = Sum_{k = 0..n} binomial(n-k-1, k)*(-1)^k*A001045(n-2k). (End)

a(n) = (A006369(n) - (A006369(n) mod 2) * (-1)^(n mod 3)) / (1 + A006369(n) mod 2). - Reinhard Zumkeller, Jan 23 2005

a(n) = A004773(n) - A004523(n). - Reinhard Zumkeller, Aug 29 2005

a(n) = floor(n/3) + ceiling(n/3). - Jonathan Vos Post, Mar 19 2006

a(n+1) = A008620(2n). - Philippe Deléham, Dec 14 2006

a(A032766(n)) = n. - Reinhard Zumkeller, Oct 30 2009

a(n) = floor((2*n^2+4*n+2)/(3*n+4)). - Gary Detlefs, Jul 13 2010

Euler transform of length 4 sequence [1, 1, 1, -1]. - Michael Somos, Jul 03 2014

a(n) = n - floor((n+1)/3). - Wesley Ivan Hurt, Sep 17 2015

a(n) = A092200(n) - floor((n+5)/3). - Filip Zaludek, Oct 27 2016

a(n) = -a(-n) for all n in Z. - Michael Somos, Oct 30 2016

E.g.f.: (2/9)*(3*exp(x)*x + sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2)). - Stefano Spezia, Sep 20 2022

Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/2. - Amiram Eldar, Sep 29 2022