A004523 - OEIS

A004523

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

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

(list; graph; refs; listen; history; text; internal format)

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

First differences are in A011655. - R. J. Mathar, Mar 19 2008

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(n-1)^2, 2n^2[. - Ralf Steiner, Jun 25 2017

a(n) is also the total domination number of the n-triangular (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 n-1, for n > 1. See Ackerman and Pinchasi. - Sela Fried, Jan 16 2023

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

E. Ackerman and R. Pinchasi, Covering a chessboard with staircase walks, Discrete Mathematics, 313 (2013).

Allan Bickle, Two Short Proofs on Total Domination, Discuss Math Graph Theory, 33 2 (2013), 457-459.

Alex Bogomolny and Don Greenwell, Static Mastermind Game, Cut The Knot!, December 1999.

R. C. Brigham, J. R. Carrington, and R. P. Vitray, Connected graphs with maximum total domination number, J. Combin. Comput. Combin. Math. 34 (2000), 81-96.

E. J. Cockayne, R. M. Dawes, and S. T. Hedetniemi, Total domination in graphs, Networks 10 (1980), 211-219.

Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint, 2016.

Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47.

Francis Laclé, 2-adic parity explorations of the 3n+ 1 problem, hal-03201180v2 [cs.DM], 2021.

G. Rosenbaum, (Static-)Mastermind.

Paul B. Slater, Formulas for Generalized Two-Qubit Separability Probabilities, arXiv:1609.08561 [quant-ph], 2016.

Paul B. Slater, Hypergeometric/Difference-Equation-Based Separability Probability Formulas and Their Asymptotics for Generalized Two-Qubit States Endowed with Random Induced Measure, arXiv:1504.04555 [quant-ph], 2015.

Eric Weisstein's World of Mathematics, Johnson Graph.

Eric Weisstein's World of Mathematics, Total Domination Number.

Eric Weisstein's World of Mathematics, Triangular Graph.

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

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(n-1)/2). - Benoit Cloitre, Nov 26 2002

a(n) = a(n-1) + (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..n-1} (Fibonacci(k) mod 2). - Paul Barry, May 31 2005

a(n) = A004773(n) - A004396(n). - Reinhard Zumkeller, Aug 29 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*(n-1)/3) = n - 1 - floor((n-1)/3). - Bruno Berselli, Jan 18 2017

a(n) = (6*n - 3 + 2*sqrt(3)*sin(2*(n-2)*Pi/3))/9. - Wesley Ivan Hurt, Sep 30 2017

Sum_{n>=2} (-1)^n/a(n) = Pi/4 (A003881). - Amiram Eldar, Sep 29 2022

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 win-loss 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 *)

Floor[2 Range[0, 20]/3] (* 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