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A001840
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Expansion of x /((1 - x)^2 * (1 - x^3)).
(Formerly M0638 N0233)
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33
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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
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OFFSET
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0,3
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COMMENTS
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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 (amarnath_murthy(AT)yahoo.com), 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]
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REFERENCES
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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.
Brian OSullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 8a, lambda=3]
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).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
Index entries for two-way infinite sequences
Index entries for sequences related to linear recurrences with constant coefficients
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.
_Simon 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.
_Simon 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
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FORMULA
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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.
floor( (n+1)*(n+2)/6 ) = floor( 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). 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]
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EXAMPLE
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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
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MAPLE
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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 (zerinvarylajos(AT)yahoo.com), 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
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MATHEMATICA
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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 *)
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PROG
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(PARI) {a(n) = (n+1) * (n+2) \ 6}
(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 (zerinvarylajos(AT)yahoo.com), Dec 01 2009]
(Haskell)
a001840 n = a001840_list !! n
a001840_list = scanl (+) 0 a008620_list
-- Reinhard Zumkeller, Apr 16 2012
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CROSSREFS
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a(n+1) = a(n) + A008620(n). - Reinhard Zumkeller, 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.
Cf. A000217, A130205.
Sequence in context: A145919 A058937 A130518 * A022794 A025693 A117930
Adjacent sequences: A001837 A001838 A001839 * A001841 A001842 A001843
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane, Neville de Mestre (neville_de_mestre(AT)macmail.bond.edu.au)
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STATUS
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approved
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