0,3

a(n-4)=number of aperiodic necklaces (Lyndon words) with 3 black beads and n-3 white beads.

Number of triangular partitions (see Almkvist).

Comment from Lekraj Beedassy (blekraj(AT)yahoo.com), Oct 13 2003: Consists of arithmetic progression quadruples of common difference n+1 starting at A045943(n). Refers to the least number of coins needed to be rearranged in order to invert the pattern of a (n+1)-rowed triangular array. For instance, a 5-rowed triangular array requires a minimum of a(4)=5 rearrangements (shown bracketed here) for it to be turned upside down.

.....{*}..................{*}*.*{*}{*}

.....*.*....................*.*.*.{*}

....*.*.*....---------\......*.*.*

..{*}*.*.*...---------/.......*.*

{*}{*}*.*{*}..................{*}

Partial sums of 1,1,1,2,2,2,3,3,3,4,4,4,... - Jon Perry (perry(AT)globalnet.co.uk), Mar 01 2004

Sum of three successive terms is a triangular number in natural order starting with 3: a(n)+a(n+1)+a(n+2) = T(n+2) = (n+2)*(n+3)/2. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 25 2004

Apply Riordan array (1/(1-x^3),x) to n. - Paul Barry (pbarry(AT)wit.ie), Apr 16 2005

In Gerhard Post et al., an opponent schedule is a team assignment S that allows n teams to play each other in 1 <= r <= n-1 rounds. The objective is to find home-away assignments for S that minimize the number of breaks (i.e. either two consecutive home or two consecutive away games of a team). A block design is used to exhibit schedules with worst-case => s(n-1)/6. Hence A001840(n) is a lower bound for opponent schedules with n teams. A general upper bound of n(n-2)/4 is established if n is of the form 4k or 4k+2, which has ceiling identical to A002620(n-1). The break minimization problem is shown to be NP-hard for each fixed r => 3. - Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 07 2007

Absolute values of numbers that appear in A145919. [From Matthew Vandermast (ghodges14(AT)comcast.net), Oct 28 2008]

In the Moree definition, (-1)^n*a(n) is the 3rd Witt transform of A033999, and (-1)^n*A004524(n) with 2 leading zeros dropped is the 2nd Witt transform of A033999. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 08 2008]

G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 73, problem 25.

R. K. Guy, A problem of Zarankiewicz, in P. Erd\"{o}s and G. Katona, editors, Theory of Graphs (Proceedings of the Colloquium, Tihany, Hungary), Academic Press, NY, 1968, pp. 119-150, (p. 126, divided by 2).

Gupta, Hansraj, Partitions of j-partite numbers into twelve or a smaller number of parts. Collection of articles dedicated to Professor P. L. Bhatnagar on his sixtieth birthday. Math. Student 40 (1972), 401-441 (1974).

Neville de Mestre and John Baker, Pebbles, Ducks and Other Surprises, Australian Maths. Teacher, Vol. 48, No 3, 1992, pp. 4-7.

Ulrich Faigle, Review of Gerhard Post and G.J. Woeginger, Sports tournaments, home-away assignments, and the break minimization problem, MR2224983(2007b:90134), 2007.

Gerhard Post and G. J. Woeginger, Sports tournaments, home-away assignments, and the break minimization problem. Discrete Optimization 3, pp. 165-173, 2006.

T. D. Noe, Table of n, a(n) for n=0..1000

G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.

D. J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 207

Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 08 2008]

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

M. Somos, Somos Polynomials

Gary E. Stevens, A Connell-Like Sequence, J. Integer Seqs., 1 (1998), #98.1.4.

Index entries for sequences related to Lyndon words

a(3k-1)=k(3k+1)/2, a(3k)=3k(k+1)/2, a(3k+1)=(k+1)(3k+2)/2.

[(n+1)(n+2)/6 ] = [A000217(n+1)/3].

G.f.: x/((1-x)^2(1-x^3)). a(n)=1+a(n-1)+a(n-3)-a(n-4), if n>3. a(-3-n)=a(n) - Michael Somos Feb 11 2004

a(n)=a(n-3)+n, a(0)=0, a(1)=1, a(2)=2. - Paul Barry (pbarry(AT)wit.ie), Jul 14 2004

a(n)=binomial(n+3, 3)/(n+3)+cos(2*pi*(n-1)/3)/9+sqrt(3)sin(2*pi*(n-1)/3)/9-1/9 - Paul Barry (pbarry(AT)wit.ie), Jan 01 2005

a(n)=sum{k=0..n, k*(cos(2*pi*(n-k)/3+pi/3)/3+sqrt(3)sin(2*pi*(n-k)/3+pi/3)/3+1/3)}; a(n)=sum{k=0..floor(n/3), n-3k}; - Paul Barry (pbarry(AT)wit.ie), Apr 16 2005

A001840 := n->floor((n+1)*(n+2)/6);

A001840:=-1/((z**2+z+1)*(z-1)**3); [Conjectured (correctly) by S. Plouffe in his 1992 dissertation.]

(PARI) a(n)=(n+1)*(n+2)\6

a(n+1) = a(n) + A008620(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 01 2002

Cf. A000031, A001037, A008748, A051168.

a(n) = (A000217(n+1)-A022003(n-1))/3 = (A016754(n+1)-A010881(A016754(n+1)))/24 = (A033996(n+1)-A010881(A033996(n+1)))/24.

Ordered union of triangular matchstick numbers A045943 and generalized pentagonal numbers A001318.

Cf. A058937.

A column of triangle A011847.

Adjacent sequences: A001837 A001838 A001839 this_sequence A001841 A001842 A001843

Sequence in context: A145919 A058937 A130518 this_sequence A130206 A022794 A025693

nonn,easy,nice

njas, Neville de Mestre (neville_de_mestre(AT)macmail.bond.edu.au)