cp's OEIS Frontend

A004523 Two even followed by one odd; or floor(2n/3).

%I A004523 #154 Feb 16 2025 08:32:28
%S A004523 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,
%T A004523 18,19,20,20,21,22,22,23,24,24,25,26,26,27,28,28,29,30,30,31,32,32,33,
%U A004523 34,34,35,36,36,37,38,38,39,40,40,41,42,42,43,44,44,45,46
%N A004523 Two even followed by one odd; or floor(2n/3).
%C A004523 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
%C A004523 First differences are in A011655. - _R. J. Mathar_, Mar 19 2008
%C A004523 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
%C A004523 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
%C A004523 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
%C A004523 a(n) is also the total domination number of the n-triangular (Johnson) graph for n > 2. - _Eric W. Weisstein_, Apr 09 2018
%C A004523 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
%C A004523 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
%H A004523 Vincenzo Librandi, <a href="/A004523/b004523.txt">Table of n, a(n) for n = 0..2000</a>
%H A004523 E. Ackerman and R. Pinchasi, <a href="https://doi.org/10.1016/j.disc.2013.07.017">Covering a chessboard with staircase walks</a>, Discrete Mathematics, 313 (2013).
%H A004523 Allan Bickle, <a href="https://doi.org/10.7151/dmgt.1655">Two Short Proofs on Total Domination</a>, Discuss Math Graph Theory, 33 2 (2013), 457-459.
%H A004523 Alex Bogomolny and Don Greenwell, <a href="http://www.cut-the-knot.org/ctk/Mastermind.shtml">Static Mastermind Game</a>, Cut The Knot!, December 1999.
%H A004523 R. C. Brigham, J. R. Carrington, and R. P. Vitray, <a href="https://www.researchgate.net/publication/268546928_Connected_graphs_with_maximum_total_domination_number">Connected graphs with maximum total domination number</a>, J. Combin. Comput. Combin. Math. 34 (2000), 81-96.
%H A004523 E. J. Cockayne, R. M. Dawes, and S. T. Hedetniemi, <a href="https://doi.org/10.1002/net.3230100304">Total domination in graphs</a>, Networks 10 (1980), 211-219.
%H A004523 Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, <a href="http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf">Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications</a>, Preprint, 2016.
%H A004523 Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, <a href="https://doi.org/10.1145/3127585">Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications</a>, ACM Transactions on Algorithms, 13:4 (2017), #47.
%H A004523 Francis Laclé, <a href="https://hal.archives-ouvertes.fr/hal-03201180v2">2-adic parity explorations of the 3n+ 1 problem</a>, hal-03201180v2 [cs.DM], 2021.
%H A004523 G. Rosenbaum, <a href="http://www.rosenbaum-games.de/3m/p1/p0071.htm">(Static-)Mastermind</a>.
%H A004523 Paul B. Slater, <a href="http://arxiv.org/abs/1609.08561">Formulas for Generalized Two-Qubit Separability Probabilities</a>, arXiv:1609.08561 [quant-ph], 2016.
%H A004523 Paul B. Slater, <a href="http://arxiv.org/abs/1504.04555">Hypergeometric/Difference-Equation-Based Separability Probability Formulas and Their Asymptotics for Generalized Two-Qubit States Endowed with Random Induced Measure</a>, arXiv:1504.04555 [quant-ph], 2015.
%H A004523 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JohnsonGraph.html">Johnson Graph</a>.
%H A004523 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TotalDominationNumber.html">Total Domination Number</a>.
%H A004523 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularGraph.html">Triangular Graph</a>.
%H A004523 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F A004523 G.f.: (x^2 + 2*x^3 + 2*x^4 + x^5)/(1 - x^3)^2, not reduced. - _Len Smiley_
%F A004523 a(n) = floor(2*n/3).
%F A004523 a(0) = a(1) = 0; for n > 1, a(n) = n - 1 - floor(a(n-1)/2). - _Benoit Cloitre_, Nov 26 2002
%F A004523 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
%F A004523 a(n) = Sum_{k=0..n-1} (Fibonacci(k) mod 2). - _Paul Barry_, May 31 2005
%F A004523 a(n) = A004773(n) - A004396(n). - _Reinhard Zumkeller_, Aug 29 2005
%F A004523 O.g.f.: x^2*(1 + x)/((1 - x)^2*(1 + x + x^2)). - _R. J. Mathar_, Mar 19 2008
%F A004523 a(n) = ceiling(2*(n-1)/3) = n - 1 - floor((n-1)/3). - _Bruno Berselli_, Jan 18 2017
%F A004523 a(n) = (6*n - 3 + 2*sqrt(3)*sin(2*(n-2)*Pi/3))/9. - _Wesley Ivan Hurt_, Sep 30 2017
%F A004523 Sum_{n>=2} (-1)^n/a(n) = Pi/4 (A003881). - _Amiram Eldar_, Sep 29 2022
%e A004523 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
%p A004523 seq(floor(2n/3), n=0..75);
%t A004523 Table[Floor[2 n/3], {n, 0, 75}]
%t A004523 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 *)
%t A004523 Floor[2 Range[0, 20]/3] (* _Eric W. Weisstein_, Apr 08 2018 *)
%t A004523 LinearRecurrence[{1, 0, 1, -1}, {0, 1, 2, 2}, {0, 20}] (* _Eric W. Weisstein_, Apr 08 2018 *)
%t A004523 CoefficientList[Series[x^2 (1 + x)/((-1 + x)^2 (1 + x + x^2)), {x, 0, 20}], x] (* _Eric W. Weisstein_, Apr 08 2018 *)
%t A004523 Table[If[EvenQ[n],{n,n},n],{n,0,50}]//Flatten (* _Harvey P. Dale_, May 27 2021 *)
%o A004523 (Haskell)
%o A004523 a004523 n = a004523_list !! n
%o A004523 a004523_list = 0 : 0 : 1 : map (+ 2) a004523_list
%o A004523 -- _Reinhard Zumkeller_, Nov 06 2012
%o A004523 (PARI) a(n)=2*n\3 \\ _Charles R Greathouse IV_, Sep 02 2015
%o A004523 (Magma) [Floor(2*n/3): n in [0..50]]; // _G. C. Greubel_, Nov 02 2017
%Y A004523 Cf. A003881, A004396, A004773, A182210, A291778, A291779.
%Y A004523 Zero followed by partial sums of A011655.
%K A004523 nonn,easy
%O A004523 0,4
%A A004523 _N. J. A. Sloane_