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A001840 Expansion of x /((1 - x)^2 * (1 - x^3)).
(Formerly M0638 N0233)
44
0, 1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 35, 40, 45, 51, 57, 63, 70, 77, 84, 92, 100, 108, 117, 126, 135, 145, 155, 165, 176, 187, 198, 210, 222, 234, 247, 260, 273, 287, 301, 315, 330, 345, 360, 376, 392, 408, 425, 442, 459, 477, 495, 513, 532, 551, 570, 590 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

Number of triangular partitions (see Almkvist).

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.

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

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

  ....*.*.*....---------\......*.*.*

  ..{*}*.*.*...---------/.......*.*

  {*}{*}*.*{*}..................{*}

  - Lekraj Beedassy, Oct 13 2003

Partial sums of 1,1,1,2,2,2,3,3,3,4,4,4,... - Jon Perry, 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, Apr 25 2004

Apply Riordan array (1/(1-x^3),x) to n. - Paul Barry, 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, Jun 07 2007

Absolute values of numbers that appear in A145919. - Matthew Vandermast, 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. - R. J. Mathar, Nov 08 2008

Column sums of:

1 2 3 4 5 6 7  8  9.....

      1 2 3 4  5  6.....

            1  2  3.....

........................

----------------------

1 2 3 5 7 9 12 15 18  - Jon Perry, Nov 16 2010

a(n) is the sum of the positive integers <= n that have the same residue modulo 3 as n. They are the additive counterpart of the triple factorial numbers. - Peter Luschny, Jul 06 2011

a(n+1) is the number of 3-tuples (w,x,y) with all terms in {0,...,n} and w=3*x+y. - Clark Kimberling, Jun 04 2012

a(n+1) is the number of pairs (x,y) with x and y in {0,...,n}, x-y = (1 mod 3), and x+y < n. - Clark Kimberling, Jul 02 2012

a(n+1) is the number of partitions of n into two sorts of part(s) 1 and one sort of (part) 3. - Joerg Arndt, Jun 10 2013

Arrange A004523 in rows successively shifted to the right two spaces and sum the columns:

1  2  2  3  4  4  5  6  6...

      1  2  2  3  4  4  5...

            1  2  2  3  4...

                  1  2  2...

                        1...

------------------------------

1  2  3  5  7  9 12 15 18... - L. Edson Jeffery, Jul 30 2014

a(n) = A258708(n+1,1) for n > 0. - Reinhard Zumkeller, Jun 23 2015

REFERENCES

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

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

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).

R. K. Guy, A problem of Zarankiewicz, in P. Erdős 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).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

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

C. Ahmed, P. Martin, V. Mazorchuk, On the number of principal ideals in d-tonal partition monoids, arXiv preprint arXiv:1503.06718 [math.CO], 2015.

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, arXiv:hep-th/9604128, 1996.

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

H. Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts, Math. Student 40 (1972), 401-441 (1974). [Annotated scanned copy]

R. K. Guy, A problem of Zarankiewicz, Research Paper No. 12, Dept. of Math., Univ. Calgary, Jan. 1967. [Annotated and scanned copy, with permission]

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.

R. P. Loh, A. G. Shannon, A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.

Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.

Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 8a, lambda=3]

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

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

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

Index entries for two-way infinite sequences

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1)

FORMULA

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.

Euler transform of length 3 sequence [ 2, 0, 1].

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

a(n) = floor( (n+1)*(n+2)/6 ) = floor( A000217(n+1)/3 ).

a(n+1) = a(n) + A008620(n). - Reinhard Zumkeller, Aug 01 2002

G.f.: x / ((1-x)^2 * (1-x^3)). a(n) = 1 + a(n-1) + a(n-3) - a(n-4). 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, 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, 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-3*k}. - Paul Barry, Apr 16 2005

a(0) = 0, a(1) = 1; for n>1, a(n) = t(n)-a(n-1)-a(n-2), where t(n) = n(n+1)/2 = A000217(n).

G.f.: x/(1+x+x^2)/(1-x)^3. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009

a(n) = (4+3*n^2+9*n)/18+((n mod 3)-((n-1) mod 3))/9. - Klaus Brockhaus, Oct 01 2009

a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5), with a(0)=0, a(1)=1, a(2)=2, a(3)=3, a(4)=5. - Harvey P. Dale, Jul 25 2011

a(n) = A214734(n + 2, 1, 3). - Renzo Benedetti, Aug 27 2012

G.f.: x*G(0), where G(k)= 1 + x*(3*k+4)/(3*k + 2 - 3*x*(k+2)*(3*k+2)/(3*(1+x)*k + 6*x + 4 - x*(3*k+4)*(3*k+5)/(x*(3*k+5) + 3*(k+1)/G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Jun 10 2013

Empirical: a(n) = floor((n+3)/(e^(6/(n+3))-1)). - Richard R. Forberg, Jul 24 2013

a(n) = sum( floor((i+2)/3), i=0..n ). - Bruno Berselli, Aug 29 2013

0 = a(n)*(a(n+2) + a(n+3)) + a(n+1)*(-2*a(n+2) - a(n+3) + a(n+4)) + a(n+2)*(a(n+2) - 2*a(n+3) + a(n+4)) for all n in Z. - Michael Somos, Jan 22 2014

EXAMPLE

G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 9*x^6 + 12*x^7 + 15*x^8 + 18*x^9 + ...

1+2+3=6=t(3), 2+3+5=t(4), 5+7+9=t(5).

[n] a(n)

--------

[1] 1

[2] 2

[3] 3

[4] 1 + 4

[5] 2 + 5

[6] 3 + 6

[7] 1 + 4 + 7

[8] 2 + 5 + 8

[9] 3 + 6 + 9

a(7) = floor(2/3) +floor(3/3) +floor(4/3) +floor(5/3) +floor(6/3) +floor(7/3) +floor(8/3) +floor(9/3) = 12. - Bruno Berselli, Aug 29 2013

MAPLE

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

A001840:=-1/((z**2+z+1)*(z-1)**3); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

seq(floor(binomial(n-1, 2)/3), n=3..61); # Zerinvary Lajos, Jan 12 2009

A001840 :=  n -> add(k, k = select(k -> k mod 3 = n mod 3, [$1 .. n])): seq(A001840(n), n = 0 .. 58); # Peter Luschny, Jul 06 2011

MATHEMATICA

a[0]=0; a[1]=1; a[n_]:= a[n]= n(n+1)/2 -a[n-1] -a[n-2]; Table[a[n], {n, 0, 100}]

f[n_] := Floor[(n + 1)(n + 2)/6]; Array[f, 59, 0] (* Or *)

CoefficientList[ Series[ x/((1 + x + x^2)*(1 - x)^3), {x, 0, 58}], x] (* Robert G. Wilson v *)

a[ n_] := With[{m = If[ n < 0, -3 - n, n]}, SeriesCoefficient[ x /((1 - x^3) (1 - x)^2), {x, 0, m}]] (* Michael Somos, Jul 11 2011 *)

LinearRecurrence[{2, -1, 1, -2, 1}, {0, 1, 2, 3, 5}, 60] (* Harvey P. Dale, Jul 25 2011 *)

PROG

(PARI) {a(n) = (n+1) * (n+2) \ 6} /* Michael Somos, Feb 11 2004 */

(MAGMA) [ n le 2 select n else n*(n+1)/2-Self(n-1)-Self(n-2): n in [1..58] ];  // Klaus Brockhaus, Oct 01 2009

(Sage) [floor(binomial(n, 2)/3) for n in xrange(3, 61)]# Zerinvary Lajos, Dec 01 2009

(Haskell)

a001840 n = a001840_list !! n

a001840_list = scanl (+) 0 a008620_list

-- Reinhard Zumkeller, Apr 16 2012

CROSSREFS

Cf. A000031, A001037, A008748, A051168.

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

Cf. A058937.

A column of triangle A011847.

Cf. A000217, A130205.

Cf. A258708.

Sequence in context: A145919 A058937 A130518 * A022794 A025693 A117930

Adjacent sequences:  A001837 A001838 A001839 * A001841 A001842 A001843

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Neville de Mestre (neville_de_mestre(AT)macmail.bond.edu.au)

STATUS

approved

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Last modified September 26 02:44 EDT 2016. Contains 276542 sequences.