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A004396 One even number followed by two odd numbers. 38
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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.

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

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

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

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

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 - From 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 to sequences related to Olympiads.

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

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

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(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(sqrt(3)*x/2)). - Stefano Spezia, Sep 20 2022

EXAMPLE

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 + ...

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 *)

PROG

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

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

(Haskell)

a004396 n = a004396_list !! n

a004396_list = 0 : 1 : 1 : map (+ 2) a004396_list

-- Reinhard Zumkeller, Nov 06 2012

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

CROSSREFS

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

Sequence in context: A156689 A168052 A131737 * A066481 A248103 A121928

Adjacent sequences:  A004393 A004394 A004395 * A004397 A004398 A004399

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified September 27 19:37 EDT 2022. Contains 357063 sequences. (Running on oeis4.)