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%I #101 Feb 28 2024 12:44:05
%S 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,
%T 15,15,15,16,17,17,17,18,19,19,19,20,21,21,21,22,23,23,23,24,25,25,25,
%U 26,27,27,27,28,29,29,29,30,31,31,31,32,33,33,33,34,35,35,35,36,37,37,37
%N One even followed by three odd.
%C 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
%C (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
%C 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
%C Also the domination number of the n X n black bishop graph. - _Eric W. Weisstein_, Jun 26 2017
%C Also the domination number of the (n-1)-Moebius laddder. - _Eric W. Weisstein_, Jun 30 2017
%C Also the rook domination number of the hexagonal hexagon board B_n [Harborth and Nienborg] - _N. J. A. Sloane_, Aug 31 2021
%C 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
%D A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 247.
%D Y. Ito, I. Nakamura, Hilbert schemes and simple singularities, New trends in algebraic geometry (Warwick, 1996), 151-233, Cambridge University Press, 1999.
%H Vincenzo Librandi, <a href="/A004525/b004525.txt">Table of n, a(n) for n = 0..10000</a>
%H Heiko Harborth and Hauke Nienborg, <a href="http://math.colgate.edu/~integers/graham14/graham14.Abstract.html">Rook domination on hexagonal hexagon boards</a>, INTEGERS 21A (2021), #A14.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BlackBishopGraph.html">Black Bishop Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DominationNumber.html">Domination Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MoebiusLadder.html">Moebius Ladder</a>
%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1).
%F a(n) = a(n-1) - a(n-2) + a(n-3) + 1 = n - A004524(n+1). - _Henry Bottomley_, Mar 08 2000
%F 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
%F a(n) = -a(-n) for all n in Z. - _Michael Somos_, Jul 19 2003
%F 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
%F a(n) = Sum_{k=0..n-1} (1 + (-1)^binomial(k+1, 2))/2. - _Paul Barry_, Mar 31 2008
%F E.g.f: A(x) = (x*exp(x) + sin(x))/2. - _Vladimir Kruchinin_, Feb 20 2011
%F a(n) = (1/4)*(2*n - (1 - (-1)^n)*(-1)^(n*(n+1)/2)). - _Bruno Berselli_, Mar 13 2012
%F a(n) = (n - floor(cos(Pi*(n+1)/2)))/2. - _Wesley Ivan Hurt_, Oct 22 2013
%F Euler transform of length 6 sequence [1, 0, 1, 1, 0, -1]. - _Michael Somos_, Apr 03 2017
%F a(n) = (n + sin(n*Pi/2))/2. - _Wesley Ivan Hurt_, Oct 02 2017
%F a(n) = n-1-a(n-2) for n >= 2. - _Kritsada Moomuang_, Oct 29 2019
%e 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 + ...
%p A004525 := proc(n): floor(n/4) + ceil(n/4) end: seq(A004525(n), n=0..75); # _Johannes W. Meijer_, Aug 06 2011
%t Table[Floor[n/4] + Ceiling[n/4], {n, 0, 100}] (* _Wesley Ivan Hurt_, Oct 22 2013 *)
%t Table[(n + Sin[n Pi/2])/2, {n, 0, 30}] (* _Eric W. Weisstein_, Jun 30 2017 *)
%t LinearRecurrence[{2, -2, 2, -1}, {1, 1, 1, 2}, {0, 20}] (* _Eric W. Weisstein_, Jun 30 2017 *)
%t Table[{n-1,n,n,n},{n,1,41,2}]//Flatten (* _Harvey P. Dale_, Oct 18 2019 *)
%o (PARI) {a(n) = n\4 + (n+3)\4}; /* _Michael Somos_, Jul 19 2003 */
%o (Magma) [Floor(n/4) + Ceiling(n/4): n in [0..70]]; // _Vincenzo Librandi_, Aug 07 2011
%o (Maxima) makelist((1/4)*(2*n-(1-(-1)^n)*(-1)^(n*(n+1)/2)), n, 0, 75); /* _Bruno Berselli_, Mar 13 2012 */
%o (Haskell)
%o a004525 n = a004525_list !! n
%o a004525_list = 0 : 1 : 1 : zipWith3 (\x y z -> x - y + z + 1)
%o a004525_list (tail a004525_list) (drop 2 a004525_list)
%o -- _Reinhard Zumkeller_, Jul 14 2012
%o (Python)
%o def A004525(n): return ((n>>1)&-2)+bool(n&3) # _Chai Wah Wu_, Jan 27 2023
%Y Cf. A002620, A004396, A004524.
%K nonn,easy
%O 0,5
%A _N. J. A. Sloane_
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