A004525 One even followed by three odd.

0, 1, 1, 1, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 7, 7, 8, 9, 9, 9, 10, 11, 11, 11, 12, 13, 13, 13, 14, 15, 15, 15, 16, 17, 17, 17, 18, 19, 19, 19, 20, 21, 21, 21, 22, 23, 23, 23, 24, 25, 25, 25, 26, 27, 27, 27, 28, 29, 29, 29, 30, 31, 31, 31, 32, 33, 33, 33, 34, 35, 35, 35, 36, 37, 37, 37

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n+1) is the composition length of the n-th symmetric power of the natural representation of a finite subgroup of SL(2,C) of type E_6 (binary tetrahedral group). - Paul Boddington, Oct 23 2003

(1 + x + x^2 + x^3 + x^4 + x^5) / ( (1-x^3)*(1- x^4)) is the Poincaré series [or Poincare series] (or Molien series) for H^*(GL_2(F_3)). - N. J. A. Sloane, Jun 12 2004

The Fi1 and Fi2 sums, see A180662 for the definition of these sums, of triangle A101950 equal the terms of this sequence without the first term. - Johannes W. Meijer, Aug 06 2011

Also the domination number of the n X n black bishop graph. - Eric W. Weisstein, Jun 26 2017

Also the domination number of the (n-1)-Moebius laddder. - Eric W. Weisstein, Jun 30 2017

Also the rook domination number of the hexagonal hexagon board B_n [Harborth and Nienborg] - N. J. A. Sloane, Aug 31 2021

Two players play a game, the object of which is to determine a score. Player 1 prefers larger scores, while player 2 prefers smaller scores. The game begins with a set of potential scores {1,2,3, ... n}. Player 1 divides this set into two nonempty sets, one of which player 2 chooses. Player 2 the divides their chosen set into two nonempty sets, one of which player 1 chooses, and so on, until the final score is arrived at. a(n+1) is the final score when both players play optimally. - Thomas Anton, Jul 14 2023

Examples

			G.f. = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + ...
		

References

• A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 247.
• Y. Ito, I. Nakamura, Hilbert schemes and simple singularities, New trends in algebraic geometry (Warwick, 1996), 151-233, Cambridge University Press, 1999.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Heiko Harborth and Hauke Nienborg, Rook domination on hexagonal hexagon boards, INTEGERS 21A (2021), #A14.
• Eric Weisstein's World of Mathematics, Black Bishop Graph
• Eric Weisstein's World of Mathematics, Domination Number
• Eric Weisstein's World of Mathematics, Moebius Ladder
• Index entries for two-way infinite sequences
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Crossrefs

Cf. A002620, A004396, A004524.

Programs

Haskell

a004525 n = a004525_list !! n
a004525_list = 0 : 1 : 1 : zipWith3 (\x y z -> x - y + z + 1)
               a004525_list (tail a004525_list) (drop 2 a004525_list)
-- Reinhard Zumkeller, Jul 14 2012

Magma

[Floor(n/4) + Ceiling(n/4): n in [0..70]]; // Vincenzo Librandi, Aug 07 2011

Maple

A004525 := proc(n): floor(n/4) + ceil(n/4) end: seq(A004525(n), n=0..75); # Johannes W. Meijer, Aug 06 2011

Mathematica

Table[Floor[n/4] + Ceiling[n/4], {n, 0, 100}] (* Wesley Ivan Hurt, Oct 22 2013 *)
Table[(n + Sin[n Pi/2])/2, {n, 0, 30}] (* Eric W. Weisstein, Jun 30 2017 *)
LinearRecurrence[{2, -2, 2, -1}, {1, 1, 1, 2}, {0, 20}] (* Eric W. Weisstein, Jun 30 2017 *)
Table[{n - 1, n, n, n}, {n, 1, 41, 2}] // Flatten (* Harvey P. Dale, Oct 18 2019 *)

Maxima

makelist((1/4)*(2*n-(1-(-1)^n)*(-1)^(n*(n+1)/2)), n, 0, 75); /* Bruno Berselli, Mar 13 2012 */

PARI

{a(n) = n\4 + (n+3)\4}; /* Michael Somos, Jul 19 2003 */

Python

def A004525(n): return ((n>>1)&-2)+bool(n&3) # Chai Wah Wu, Jan 27 2023

Formula

a(n) = a(n-1) - a(n-2) + a(n-3) + 1 = n - A004524(n+1). - Henry Bottomley, Mar 08 2000

G.f.: x*(1-x+x^2)/((1-x)^2*(1+x^2)) = x*(1-x^6)/((1-x)*(1-x^3)*(1-x^4)). - Michael Somos, Jul 19 2003

a(n) = -a(-n) for all n in Z. - Michael Somos, Jul 19 2003

a(n) = floor(n/4) + ceiling(n/4). See also A004396, one even followed by two odd and A002620, quarter-squares: floor(n/2)*ceiling(n/2). - Jonathan Vos Post, Mar 19 2006

a(n) = Sum_{k=0..n-1} (1 + (-1)^binomial(k+1, 2))/2. - Paul Barry, Mar 31 2008

E.g.f: A(x) = (x*exp(x) + sin(x))/2. - Vladimir Kruchinin, Feb 20 2011

a(n) = (1/4)*(2*n - (1 - (-1)^n)*(-1)^(n*(n+1)/2)). - Bruno Berselli, Mar 13 2012

a(n) = (n - floor(cos(Pi*(n+1)/2)))/2. - Wesley Ivan Hurt, Oct 22 2013

Euler transform of length 6 sequence [1, 0, 1, 1, 0, -1]. - Michael Somos, Apr 03 2017

a(n) = (n + sin(n*Pi/2))/2. - Wesley Ivan Hurt, Oct 02 2017

a(n) = n-1-a(n-2) for n >= 2. - Kritsada Moomuang, Oct 29 2019