%I #278 Aug 15 2024 10:59:15

%S 0,1,3,4,6,7,9,10,12,13,15,16,18,19,21,22,24,25,27,28,30,31,33,34,36,

%T 37,39,40,42,43,45,46,48,49,51,52,54,55,57,58,60,61,63,64,66,67,69,70,

%U 72,73,75,76,78,79,81,82,84,85,87,88,90,91,93,94,96,97,99,100,102,103

%N Numbers that are congruent to 0 or 1 (mod 3).

%C Omitting the initial 0, a(n) is the number of 1's in the n-th row of the triangle in A118111. - _Hans Havermann_, May 26 2002

%C Binomial transform is A053220. - _Michael Somos_, Jul 10 2003

%C Smallest number of different people in a set of n-1 photographs that satisfies the following conditions: In each photograph there are 3 women, the woman in the middle is the mother of the person on her left and is a sister of the person on her right and the women in the middle of the photographs are all different. - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Sep 22 2006

%C Partial sums of A000034. - _Richard Choulet_, Jan 28 2010

%C Starting with 1 = row sums of triangle A171370. - _Gary W. Adamson_, Feb 15 2010

%C a(n) is the set of values for m in which 6k + m can be a perfect square (quadratic residues of 6 including trivial case of 0). - _Gary Detlefs_, Mar 19 2010

%C For n >= 2, a(n) is the smallest number with n as an anti-divisor. - _Franklin T. Adams-Watters_, Oct 28 2011

%C Sequence is also the maximum number of floors with 3 elevators and n stops in a "Convenient Building". See A196592 and Erich Friedman link below. - _Robert Price_, May 30 2013

%C a(n) is also the total number of coins left after packing 4-curves patterns (4c2) into a fountain of coins base n. The total number of 4c2 is A002620 and voids left is A000982. See illustration in links. - _Kival Ngaokrajang_, Oct 26 2013

%C Number of partitions of 6n into two even parts. - _Wesley Ivan Hurt_, Nov 15 2014

%C Number of partitions of 3n into exactly 2 parts. - _Colin Barker_, Mar 23 2015

%C Nonnegative m such that floor(2*m/3) = 2*floor(m/3). - _Bruno Berselli_, Dec 09 2015

%C For n >= 3, also the independence number of the n-web graph. - _Eric W. Weisstein_, Dec 31 2015

%C Equivalently, nonnegative numbers m for which m*(m+2)/3 and m*(m+5)/6 are integers. - _Bruno Berselli_, Jul 18 2016

%C Also the clique covering number of the n-Andrásfai graph for n > 0. - _Eric W. Weisstein_, Mar 26 2018

%C Maximum sum of degeneracies over all decompositions of the complete graph of order n+1 into three factors. The extremal decompositions are characterized in the Bickle link below. - _Allan Bickle_, Dec 21 2021

%C Also the Hadwiger number of the n-cocktail party graph. - _Eric W. Weisstein_, Apr 30 2022

%H Indranil Ghosh, <a href="/A032766/b032766.txt">Table of n, a(n) for n = 0..10000</a>

%H Allan Bickle, <a href="https://allanbickle.files.wordpress.com/2016/05/ngcorepaper.pdf">Nordhaus-Gaddum Theorems for k-Decompositions</a>, Congr. Num. 211 (2012) 171-183.

%H F. Javier de Vega, <a href="https://arxiv.org/abs/2003.13378">An extension of Furstenberg's theorem of the infinitude of primes</a>, arXiv:2003.13378 [math.NT], 2020.

%H Erich Friedman, <a href="https://erich-friedman.github.io/mathmagic/1109.html">Problem of the month November 2009</a>

%H Z. Füredi, A. Kostochka, M. Stiebitz, R. Skrekovski, and D. West, <a href="https://faculty.math.illinois.edu/~west/pubs/norgad.pdf">Nordhaus-Gaddum-type theorems for decompositions into many parts</a>, J. Graph Theory 50 (2005), 273-292.

%H Andreas M. Hinz, Sandi Klavžar, Uroš Milutinović and Ciril Petr, <a href="http://dx.doi.org/10.1007/978-3-0348-0237-6">The Tower of Hanoi - Myths and Maths</a>, Birkhäuser 2013. See page 282. [<a href="http://tohbook.info">Book's website</a>]

%H 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 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; DOI: 10.1145/3127585.

%H International Mathematical Olympiad 2001, <a href="http://www.mathdb.org/resource_sharing/prelim/se_00-01_sol.pdf">Hong Kong Preliminary Selection Contest</a>, Problem #20. [Broken link; <a href="/A032766/a032766_1.pdf">Cached copy</a>]

%H Matroids Matheplanet, <a href="http://matheplanet.de/matheplanet/nuke/html/viewtopic.php?topic=47877&amp;post_id=1519508">Number of d-generator groups of order 2^(d+1) and exponent-p class 2</a> (in German).

%H Emanuele Munarini, <a href="https://doi.org/10.4418/2021.76.1.14">Topological indices for the antiregular graphs</a>, Le Mathematiche, Vol. 76, No. 1 (2021), pp. 277-310, see p. 302.

%H Kival Ngaokrajang, <a href="/A032766/a032766.pdf">Illustration of initial terms (U)</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AndrasfaiGraph.html">Andrásfai Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CliqueCoveringNumber.html">Clique Covering Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HadwigerNumber.html">Hadwiger Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependenceNumber.html">Independence Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WebGraph.html">Web Graph</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

%F G.f.: x*(1+2*x)/((1-x)*(1-x^2)).

%F a(-n) = -A007494(n).

%F a(n) = A049615(n, 2), for n > 2.

%F From _Paul Barry_, Sep 04 2003: (Start)

%F a(n) = (6n - 1 + (-1)^n)/4.

%F a(n) = floor((3n + 2)/2) - 1 = A001651(n) - 1.

%F a(n) = sqrt(2) * sqrt( (6n-1) (-1)^n + 18n^2 - 6n + 1 )/4.

%F a(n) = Sum_{k=0..n} 3/2 - 2*0^k + (-1)^k/2. (End)

%F a(n) = 3*floor(n/2) + (n mod 2) = A007494(n) - A000035(n). - _Reinhard Zumkeller_, Apr 04 2005

%F a(n) = 2 * A004526(n) + A004526(n+1). - _Philippe Deléham_, Aug 07 2006

%F a(n) = 1 + ceiling(3*(n-1)/2). - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Sep 22 2006

%F Row sums of triangle A133083. - _Gary W. Adamson_, Sep 08 2007

%F a(n) = (cos(Pi*n) - 1)/4 + 3*n/2. - Bart Snapp (snapp(AT)coastal.edu), Sep 18 2008

%F A004396(a(n)) = n. - _Reinhard Zumkeller_, Oct 30 2009

%F a(n) = floor(n/2) + n. - _Gary Detlefs_, Mar 19 2010

%F a(n) = 3n - a(n-1) - 2, for n>0, a(0)=0. - _Vincenzo Librandi_, Nov 19 2010

%F a(n) = n + (n-1) - (n-2) + (n-3) - ... 1 = A052928(n) + A008619(n-1). - _Jaroslav Krizek_, Mar 22 2011

%F a(n) = a(n-1) + a(n-2) - a(n-3). - _Robert G. Wilson v_, Mar 28 2011

%F a(n) = Sum_{k>=0} A030308(n,k) * A003945(k). - _Philippe Deléham_, Oct 17 2011

%F a(n) = 2n - ceiling(n/2). - _Wesley Ivan Hurt_, Oct 25 2013

%F a(n) = A000217(n) - 2 * A002620(n-1). - _Kival Ngaokrajang_, Oct 26 2013

%F a(n) = Sum_{i=1..n} gcd(i, 2). - _Wesley Ivan Hurt_, Jan 23 2014

%F a(n) = 2n + floor((-n - (n mod 2))/2). - _Wesley Ivan Hurt_, Mar 31 2014

%F A092942(a(n)) = n for n > 0. - _Reinhard Zumkeller_, Dec 13 2014

%F a(n) = floor(3*n/2). - _L. Edson Jeffery_, Jan 18 2015

%F a(n) = A254049(A249745(n)) = (1+A007310(n)) / 2 for n >= 1. - _Antti Karttunen_, Jan 24 2015

%F E.g.f.: (3*x*exp(x) - sinh(x))/2. - _Ilya Gutkovskiy_, Jul 18 2016

%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(6*sqrt(3)) + log(3)/2. - _Amiram Eldar_, Dec 04 2021

%p a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=a[n-2]+3 od: seq(a[n], n=0..69); # _Zerinvary Lajos_, Mar 16 2008

%p seq(floor(n/2)+n, n=0..69); # _Gary Detlefs_, Mar 19 2010

%p select(n->member(n mod 3,{0,1}), [$0..103]); # _Peter Luschny_, Apr 06 2014

%t a[n_] := a[n] = 2a[n - 1] - 2a[n - 3] + a[n - 4]; a[0] = 0; a[1] = 1; a[2] = 3; a[3] = 4; Array[a, 60, 0] (* _Robert G. Wilson v_, Mar 28 2011 *)

%t Select[Range[0, 200], MemberQ[{0, 1}, Mod[#, 3]] &] (* _Vladimir Joseph Stephan Orlovsky_, Feb 11 2012 *)

%t Flatten[{#,#+1}&/@(3Range[0,40])] (* or *) LinearRecurrence[{1,1,-1}, {0,1,3}, 100] (* or *) With[{nn=110}, Complement[Range[0,nn], Range[2,nn,3]]] (* _Harvey P. Dale_, Mar 10 2013 *)

%t CoefficientList[Series[x (1 + 2 x) / ((1 - x) (1 - x^2)), {x, 0, 100}], x] (* _Vincenzo Librandi_, Nov 16 2014 *)

%t Floor[3 Range[0, 69]/2] (* _L. Edson Jeffery_, Jan 14 2017 *)

%t Drop[Range[0,110],{3,-1,3}] (* _Harvey P. Dale_, Sep 02 2023 *)

%o (PARI) {a(n) = n + n\2}

%o (Magma) &cat[ [n, n+1]: n in [0..100 by 3] ]; // _Vincenzo Librandi_, Nov 16 2014

%o (Haskell)

%o a032766 n = div n 2 + n -- _Reinhard Zumkeller_, Dec 13 2014

%o (MIT/GNU Scheme) (define (A032766 n) (+ n (floor->exact (/ n 2)))) ;; _Antti Karttunen_, Jan 24 2015

%o (PARI) concat(0, Vec(x*(1+2*x)/((1-x)*(1-x^2)) + O(x^100))) \\ _Altug Alkan_, Dec 09 2015

%o (SageMath) [int(3*n//2) for n in range(101)] # _G. C. Greubel_, Jun 23 2024

%Y Cf. A006578 (partial sums), A000034 (first differences), A016789 (complement).

%Y Essentially the same: A049624.

%Y Column 1 (the second leftmost) of triangular table A026374.

%Y Column 1 (the leftmost) of square array A191450.

%Y Row 1 of A254051.

%Y Row sums of A171370.

%Y Cf. A066272 for anti-divisors.

%Y Cf. A253888 and A254049 (permutations of this sequence without the initial zero).

%Y Cf. A254103 and A254104 (pair of permutations based on this sequence and its complement).

%Y Cf. A000035, A000217, A000982, A001651, A002620, A002717, A003945.

%Y Cf. A004396, A004526, A007310, A007494, A008619, A030308, A035360.

%Y Cf. A047270, A049615, A052928, A053220, A070893, A084056, A092942.

%Y Cf. A118111, A132463, A133083, A196592, A249745.

%K nonn,easy,nice

%O 0,3

%A _Patrick De Geest_, May 15 1998

%E Better description from _N. J. A. Sloane_, Aug 01 1998