A032766 - OEIS

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(2*m/3) = 2*floor(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-Andrásfai 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. Füredi, 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, Andrásfai 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+2*x)/((1-x)*(1-x^2)).

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

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

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 - 2*0^k + (-1)^k/2. (End)

a(n) = 3*floor(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(Pi*n) - 1)/4 + 3*n/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(3*n/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.: (3*x*exp(x) - sinh(x))/2. - Ilya Gutkovskiy, Jul 18 2016

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(6*sqrt(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 *)

Drop[Range[0, 110], {3, -1, 3}] (* Harvey P. Dale, Sep 02 2023 *)

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+2*x)/((1-x)*(1-x^2)) + O(x^100))) \\ Altug Alkan, Dec 09 2015

(SageMath) [int(3*n//2) for n in range(101)] # G. C. Greubel, Jun 23 2024

CROSSREFS