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A004525
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One even followed by three odd.
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25
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,5
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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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) = 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
a(n) = (1/4)*(2*n - (1 - (-1)^n)*(-1)^(n*(n+1)/2)). - Bruno Berselli, Mar 13 2012
Euler transform of length 6 sequence [1, 0, 1, 1, 0, -1]. - Michael Somos, Apr 03 2017
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EXAMPLE
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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 + ...
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MAPLE
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MATHEMATICA
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Table[Floor[n/4] + Ceiling[n/4], {n, 0, 100}] (* Wesley Ivan Hurt, Oct 22 2013 *)
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 *)
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PROG
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(Maxima) makelist((1/4)*(2*n-(1-(-1)^n)*(-1)^(n*(n+1)/2)), n, 0, 75); /* Bruno Berselli, Mar 13 2012 */
(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)
(Python)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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