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A004523 Two even followed by one odd; or floor(2n/3). 63
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 [seems to be INCORRECT, someone please check!]

a(n) is also the total domination number of the n-trianguar (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

REFERENCES

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.

Hsien-Kuei Hwang, S Janson, TH 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; http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf. Also Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585

LINKS

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

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.

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

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) = 2*(-1 + Sum_{k=0..n} (1/9*(-2*(k mod 3) + ((k+1) mod 3) + 4*((k+2) mod 3))) + (((n+2) mod 3) mod 2). - Paolo P. Lava, Oct 02 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

a004523_list = 0 : 0 : 1 : map (+ 2) a004523_list

-- Reinhard Zumkeller, Nov 06 2012

(PARI) a(n)=2*n\3 \\ Charles R Greathouse IV, Sep 02 2015

(Magma) [Floor(2*n/3): n in [0..50]]; // G. C. Greubel, Nov 02 2017

CROSSREFS

Cf. A003881, A004396, A004773, A291778, A291779.

Zero followed by partial sums of A011655.

Sequence in context: A195124 A032509 A322042 * A038372 A121930 A020909

Adjacent sequences: A004520 A004521 A004522 * A004524 A004525 A004526

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Dead link fixed by Nathaniel Johnston, Sep 20 2012

STATUS

approved

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Last modified December 7 15:01 EST 2022. Contains 358667 sequences. (Running on oeis4.)