

A004523


Two even followed by one odd; or floor(2n/3).


65



0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 18, 18, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 30, 31, 32, 32, 33, 34, 34, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 44, 44, 45, 46
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OFFSET

0,4


COMMENTS

Guenther Rosenbaum showed that the sequence represents the optimal number of guesses in the static Mastermind game with two pegs. Namely, the optimal number of static guesses equals 2k, if the number of colors is either (3k  1) or 3k and is (2k + 1), if the number of colors is (3k + 1), k >= 1.  Alex Bogomolny, Mar 06 2002
a(n+1) is the maximum number of wins by a team in a sequence of n basketball games if the team's longest winning streak is 2 games. See example below. In general, floor(k(n+1)/(k+1)) gives the maximum number of wins in n games when the longest winning streak is of length k.  Dennis P. Walsh, Apr 18 2012
Sum_{n>=2} 1/a(n)^k = Sum_{j>=1} Sum_{i=1..2} 1/(i*j)^k = Zeta(k)^2  Zeta(k)*Zeta(k,3), where Zeta(,) is the generalized Riemann zeta function, for the case k=2 this sum is 5*Pi^2/24.  Enrique Pérez Herrero, Jun 25 2012
a(n) is the pattern of (0+2k, 0+2k, 1+2k), k>=0. a(n) is also the number of odd integers divisible by 3 in ]2(n1)^2, 2n^2[.  Ralf Steiner, Jun 25 2017
a(n) is also the total domination number of the ntriangular (Johnson) graph for n > 2.  Eric W. Weisstein, Apr 09 2018
a(n) is the maximum total domination number of connected graphs with order n>2. The extremal graphs are "brushes", as defined in the links below.  Allan Bickle, Dec 24 2021
a(n) is the minimal number of ascending or descending staircase walks necessary to cover a chessboard of size n1, for n > 1. See Ackerman and Pinchasi.  Sela Fried, Jan 16 2023


LINKS



FORMULA

G.f.: (x^2 + 2*x^3 + 2*x^4 + x^5)/(1  x^3)^2, not reduced.  Len Smiley
a(n) = floor(2*n/3).
a(0) = a(1) = 0; for n > 1, a(n) = n  1  floor(a(n1)/2).  Benoit Cloitre, Nov 26 2002
a(n) = a(n1) + (1/2)*((1)^floor((2*n+2)/3)+1), with a(0)=0.  Mario Catalani (mario.catalani(AT)unito.it), Oct 20 2003
a(n) = Sum_{k=0..n1} (Fibonacci(k) mod 2).  Paul Barry, May 31 2005
O.g.f.: x^2*(1 + x)/((1  x)^2*(1 + x + x^2)).  R. J. Mathar, Mar 19 2008
a(n) = ceiling(2*(n1)/3) = n  1  floor((n1)/3).  Bruno Berselli, Jan 18 2017
a(n) = (6*n  3 + 2*sqrt(3)*sin(2*(n2)*Pi/3))/9.  Wesley Ivan Hurt, Sep 30 2017


EXAMPLE

For n=11, we have a(11)=7 since there are at most 7 wins by a team in a sequence of 10 games in which its longest winning streak is 2 games. One such winloss sequence with 7 wins is wwlwwlwwlw.  Dennis P. Walsh, Apr 18 2012


MAPLE

seq(floor(2n/3), n=0..75);


MATHEMATICA

Table[Floor[2 n/3], {n, 0, 75}]
Table[(6 n + 3 Cos[2 n Pi/3]  Sqrt[3] Sin[2 n Pi/3]  3)/9, {n, 0, 20}] (* Eric W. Weisstein, Apr 08 2018 *)
LinearRecurrence[{1, 0, 1, 1}, {0, 1, 2, 2}, {0, 20}] (* Eric W. Weisstein, Apr 08 2018 *)
CoefficientList[Series[x^2 (1 + x)/((1 + x)^2 (1 + x + x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Apr 08 2018 *)
Table[If[EvenQ[n], {n, n}, n], {n, 0, 50}]//Flatten (* Harvey P. Dale, May 27 2021 *)


PROG

(Haskell)
a004523 n = a004523_list !! n
a004523_list = 0 : 0 : 1 : map (+ 2) a004523_list
(Magma) [Floor(2*n/3): n in [0..50]]; // G. C. Greubel, Nov 02 2017


CROSSREFS

Zero followed by partial sums of A011655.


KEYWORD

nonn,easy


AUTHOR



STATUS

approved



