A032766 - OEIS

A032766

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

107

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, 37, 39, 40, 42, 43, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 75, 76, 78, 79, 81, 82, 84, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100, 102, 103 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,3`
	COMMENTS	`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` `Binomial transform is A053220. - Michael Somos, Jul 10 2003` `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` `Partial sums of A000034. - Richard Choulet, Jan 28 2010` `Starting with 1 = row sums of triangle A171370. - Gary W. Adamson, Feb 15 2010` `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` `For n >= 2, a(n) is the smallest number with n as an anti-divisor. - Franklin T. Adams-Watters, Oct 28 2011` `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` `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` `Number of partitions of 6n into two even parts. - Wesley Ivan Hurt, Nov 15 2014` `Number of partitions of 3n into exactly 2 parts. - Colin Barker, Mar 23 2015` `Nonnegative m such that floor(2m/3) = 2floor(m/3). - Bruno Berselli, Dec 09 2015` `For n >= 3, also the independence number of the n-web graph. - Eric W. Weisstein, Dec 31 2015` `Equivalently, nonnegative numbers m for which m(m+2)/3 and m(m+5)/6 are integers. - Bruno Berselli, Jul 18 2016` `Also the clique covering number of the n-Andrasfai graph for n > 0. - Eric W. Weisstein, Mar 26 2018` `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` `Also the Hadwiger number of the n-cocktail party graph. - Eric W. Weisstein, Apr 30 2022`
	LINKS	`Indranil Ghosh, Table of n, a(n) for n = 0..10000` `Allan Bickle, Nordhaus-Gaddum Theorems for k-Decompositions, Congr. Num. 211 (2012) 171-183.` `F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.` `Erich Friedman, Problem of the month November 2009` `Z. Furedi, A. Kostochka, M. Stiebitz, R. Skrekovski, and D. West, Nordhaus-Gaddum-type theorems for decompositions into many parts, J. Graph Theory 50 (2005), 273-292.` `Andreas M. Hinz, Sandi Klavžar, Uroš Milutinović and Ciril Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 282. [Book's website]` `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; DOI: 10.1145/3127585.` `International Mathematical Olympiad 2001, Hong Kong Preliminary Selection Contest, Problem #20. [Broken link; Cached copy]` `Matroids Matheplanet, Number of d-generator groups of order 2^(d+1) and exponent-p class 2 (in German).` `Emanuele Munarini, Topological indices for the antiregular graphs, Le Mathematiche, Vol. 76, No. 1 (2021), pp. 277-310, see p. 302.` `Kival Ngaokrajang, Illustration of initial terms (U).` `Eric Weisstein's World of Mathematics, Andrasfai Graph` `Eric Weisstein's World of Mathematics, Clique Covering Number` `Eric Weisstein's World of Mathematics, Cocktail Party Graph` `Eric Weisstein's World of Mathematics, Hadwiger Number` `Eric Weisstein's World of Mathematics, Independence Number` `Eric Weisstein's World of Mathematics, Web Graph` `Index entries for linear recurrences with constant coefficients, signature (1,1,-1).`
	FORMULA	`G.f.: x(1+2x)/((1-x)(1-x^2)).` `a(-n) = -A007494(n).` `From Paul Barry, Sep 04 2003: (Start)` `a(n) = (6n - 1 + (-1)^n)/4.` `a(n) = floor((3n + 2)/2) - 1 = A001651(n) - 1.` `a(n) = sqrt(2) sqrt( (6n-1) (-1)^n + 18n^2 - 6n + 1 )/4.` `a(n) = Sum_{k=0..n} 3/2 - 20^k + (-1)^k/2. (End)` `a(n) = 3floor(n/2) + (n mod 2) = A007494(n) - A000035(n). - Reinhard Zumkeller, Apr 04 2005` `a(n) = 2 * A004526(n) + A004526(n+1). - Philippe Deléham, Aug 07 2006` `a(n) = 1 + ceiling(3(n-1)/2). - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Sep 22 2006` `Row sums of triangle A133083. - Gary W. Adamson, Sep 08 2007` `a(n) = (cos(Pin) - 1)/4 + 3n/2. - Bart Snapp (snapp(AT)coastal.edu), Sep 18 2008` `A004396(a(n)) = n. - Reinhard Zumkeller, Oct 30 2009` `a(n) = floor(n/2) + n. - Gary Detlefs, Mar 19 2010` `a(n) = 3n - a(n-1) - 2, for n>0, a(0)=0. - Vincenzo Librandi, Nov 19 2010` `a(n) = n + (n-1) - (n-2) + (n-3) - ... 1 = A052928(n) + A008619(n-1). - Jaroslav Krizek, Mar 22 2011` `a(n) = a(n-1) + a(n-2) - a(n-3). - Robert G. Wilson v, Mar 28 2011` `a(n) = Sum_{k>=0} A030308(n,k) A003945(k). - Philippe Deléham, Oct 17 2011` `a(n) = 2n - ceiling(n/2). - Wesley Ivan Hurt, Oct 25 2013` `a(n) = A000217(n) - 2 * A002620(n-1). - Kival Ngaokrajang, Oct 26 2013` `a(n) = Sum_{i=1..n} gcd(i, 2). - Wesley Ivan Hurt, Jan 23 2014` `a(n) = 2n + floor((-n - (n mod 2))/2). - Wesley Ivan Hurt, Mar 31 2014` `A092942(a(n)) = n for n > 0. - Reinhard Zumkeller, Dec 13 2014` `a(n) = floor(3n/2). - L. Edson Jeffery, Jan 18 2015` `a(n) = A254049(A249745(n)) = (1+A007310(n)) / 2 for n >= 1. - Antti Karttunen, Jan 24 2015` `E.g.f.: (3xexp(x) - sinh(x))/2. - Ilya Gutkovskiy, Jul 18 2016` `Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(6sqrt(3)) + log(3)/2. - Amiram Eldar, Dec 04 2021`
	MAPLE	`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` `seq(floor(n/2)+n, n=0..69); # Gary Detlefs, Mar 19 2010` `select(n->member(n mod 3, {0, 1}), [$0..103]); # Peter Luschny, Apr 06 2014`
	MATHEMATICA	`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 )` `Select[Range[0, 200], MemberQ[{0, 1}, Mod[#, 3]] &] ( Vladimir Joseph Stephan Orlovsky, Feb 11 2012 )` `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 )` `CoefficientList[Series[x (1 + 2 x) / ((1 - x) (1 - x^2)), {x, 0, 100}], x] ( Vincenzo Librandi, Nov 16 2014 )` `Floor[3 Range[0, 69]/2] ( L. Edson Jeffery, Jan 14 2017 *)`
	PROG	`(PARI) {a(n) = n + n\2}` `(Magma) &cat[ [n, n+1]: n in [0..100 by 3] ]; // Vincenzo Librandi, Nov 16 2014` `(Haskell)` `a032766 n = div n 2 + n -- Reinhard Zumkeller, Dec 13 2014` `(MIT/GNU Scheme) (define (A032766 n) (+ n (floor->exact (/ n 2)))) ;; Antti Karttunen, Jan 24 2015` `(PARI) concat(0, Vec(x(1+2x)/((1-x)*(1-x^2)) + O(x^100))) \\ Altug Alkan, Dec 09 2015`
	CROSSREFS	`Complement: A016789. Essentially the same: A049624.` `For n>0, a(n) = T(n,2), array T as in A049615.` `Column 1 (the second leftmost) of triangular table A026374. Column 1 (the leftmost) of square array A191450. Row 1 of A254051.` `Partial sums are A006578. Partial sums of A000034.` `Cf. A001651, A002717, A007310, A007494, A035360, A047270, A070893, A084056, A132463, A133083, A171370.` `Cf. A066272 for anti-divisors.` `Cf. A253888 and A254049 (permutations of this sequence without the initial zero).` `Cf. A254103 and A254104 (pair of permutations based on this sequence and its complement).` `Sequence in context: A026322 A049624 A084056 * A189935 A329987 A329962` `Adjacent sequences: A032763 A032764 A032765 * A032767 A032768 A032769`
	KEYWORD	`nonn,easy,nice`
	AUTHOR	`Patrick De Geest, May 15 1998`
	EXTENSIONS	`Better description from N. J. A. Sloane, Aug 01 1998`
	STATUS	`approved`