

A004525


One even followed by three odd.


25



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
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OFFSET

0,5


COMMENTS

a(n+1) is the composition length of the nth 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) / ( (1x^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 (n1)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


REFERENCES

A. Adem and R. J. Milgram, Cohomology of Finite Groups, SpringerVerlag, 2nd. ed., 2004; p. 247.
Y. Ito, I. Nakamura, Hilbert schemes and simple singularities, New trends in algebraic geometry (Warwick, 1996), 151233, Cambridge University Press, 1999.


LINKS



FORMULA

G.f.: x*(1x+x^2)/((1x)^2*(1+x^2)) = x*(1x^6)/((1x)*(1x^3)*(1x^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, quartersquares: floor(n/2)*ceiling(n/2).  Jonathan Vos Post, Mar 19 2006
a(n) = Sum_{k=0..n1} (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


EXAMPLE

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 + ...


MAPLE



MATHEMATICA

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[{n1, n, n, n}, {n, 1, 41, 2}]//Flatten (* Harvey P. Dale, Oct 18 2019 *)


PROG

(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)


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



