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A004525 One even followed by three odd. 13
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)
OFFSET

0,5

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

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

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).

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

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

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 *)

PROG

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

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

(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)

-- Reinhard Zumkeller, Jul 14 2012

CROSSREFS

Cf. A002620, A004396, A004524.

Sequence in context: A287355 A194171 * A049206 A194247 A084767 A137580

Adjacent sequences:  A004522 A004523 A004524 * A004526 A004527 A004528

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 23 13:27 EST 2018. Contains 299581 sequences. (Running on oeis4.)