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%I M0638 N0233 #227 Apr 05 2023 18:51:10
%S 0,1,2,3,5,7,9,12,15,18,22,26,30,35,40,45,51,57,63,70,77,84,92,100,
%T 108,117,126,135,145,155,165,176,187,198,210,222,234,247,260,273,287,
%U 301,315,330,345,360,376,392,408,425,442,459,477,495,513,532,551,570,590
%N Expansion of g.f. x/((1 - x)^2*(1 - x^3)).
%C a(n-3) is the number of aperiodic necklaces (Lyndon words) with 3 black beads and n-3 white beads.
%C Number of triangular partitions (see Almkvist).
%C 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.
%C .....{*}..................{*}*.*{*}{*}
%C .....*.*....................*.*.*.{*}
%C ....*.*.*....---------\......*.*.*
%C ..{*}*.*.*...---------/.......*.*
%C {*}{*}*.*{*}..................{*}
%C - _Lekraj Beedassy_, Oct 13 2003
%C Partial sums of 1,1,1,2,2,2,3,3,3,4,4,4,... - _Jon Perry_, Mar 01 2004
%C 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
%C Apply Riordan array (1/(1-x^3),x) to n. - _Paul Barry_, Apr 16 2005
%C Absolute values of numbers that appear in A145919. - _Matthew Vandermast_, Oct 28 2008
%C 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
%C Column sums of:
%C 1 2 3 4 5 6 7 8 9.....
%C 1 2 3 4 5 6.....
%C 1 2 3.....
%C ........................
%C ----------------------
%C 1 2 3 5 7 9 12 15 18 - _Jon Perry_, Nov 16 2010
%C 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
%C 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
%C 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
%C 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
%C Arrange A004523 in rows successively shifted to the right two spaces and sum the columns:
%C 1 2 2 3 4 4 5 6 6...
%C 1 2 2 3 4 4 5...
%C 1 2 2 3 4...
%C 1 2 2...
%C 1...
%C ------------------------------
%C 1 2 3 5 7 9 12 15 18... - _L. Edson Jeffery_, Jul 30 2014
%C a(n) = A258708(n+1,1) for n > 0. - _Reinhard Zumkeller_, Jun 23 2015
%C Also the number of triples of positive integers summing to n + 4, the first less than each of the other two. Also the number of triples of positive integers summing to n + 2, the first less than or equal to each of the other two. - _Gus Wiseman_, Oct 11 2020
%D Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 73, problem 25.
%D Ulrich Faigle, Review of Gerhard Post and G.J. Woeginger, Sports tournaments, home-away assignments and the break minimization problem, MR2224983(2007b:90134), 2007.
%D Hansraj Gupta, 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).
%D Richard 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).
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001840/b001840.txt">Table of n, a(n) for n = 0..1000</a>
%H C. Ahmed, P. Martin, and V. Mazorchuk, <a href="http://arxiv.org/abs/1503.06718">On the number of principal ideals in d-tonal partition monoids</a>, arXiv preprint arXiv:1503.06718 [math.CO], 2015.
%H G. Almkvist, <a href="http://projecteuclid.org/euclid.em/1047674152">Asymptotic formulas and generalized Dedekind sums</a>, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
%H D. J. Broadhurst, <a href="http://arXiv.org/abs/hep-th/9604128">On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory</a>, arXiv:hep-th/9604128, 1996.
%H Neville de Mestre and John Baker, <a href="http://www.aamt.edu.au/Publications-and-statements/Journals/Journals-Index/The-Australian-Mathematics-Teacher/AMT-48-3-4">Pebbles, Ducks and Other Surprises</a>, Australian Maths. Teacher, Vol. 48, No 3, 1992, pp. 4-7.
%H Peter M. Chema, <a href="/A001840/a001840_2.pdf">Illustration of first 27 terms as corners of a double hexagon spiral from 0</a>
%H H. Gupta, <a href="/A001840/a001840.pdf">Partitions of j-partite numbers into twelve or a smaller number of parts</a>, Math. Student 40 (1972), 401-441 (1974). [Annotated scanned copy]
%H Richard K. Guy, <a href="/A001197/a001197.pdf">A problem of Zarankiewicz</a>, Research Paper No. 12, Dept. of Math., Univ. Calgary, Jan. 1967. [Annotated and scanned copy, with permission]
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=207">Encyclopedia of Combinatorial Structures 207</a>.
%H Clark Kimberling, <a href="https://www.emis.de/journals/JIS/VOL22/Kimberling/kimb9.html">A Combinatorial Classification of Triangle Centers on the Line at Infinity</a>, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
%H Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Kimberling/kimber67.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%H R. P. Loh, A. G. Shannon, and A. F. Horadam, <a href="/A000969/a000969.pdf">Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients</a>, Preprint, 1980.
%H Pieter Moree, <a href="http://dx.doi.org/10.1016/j.disc.2005.03.004">The formal series Witt transform</a>, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
%H Brian O'Sullivan and Thomas Busch, <a href="http://arxiv.org/abs/0810.0231">Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas</a>, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 8a, lambda=3]
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H Gerhard Post and G. J. Woeginger, <a href="http://dx.doi.org/10.1016/j.disopt.2005.08.009">Sports tournaments, home-away assignments and the break minimization problem</a>, Discrete Optimization 3, pp. 165-173, 2006.
%H Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/somospol.html">Somos Polynomials</a>.
%H Gary E. Stevens, <a href="https://cs.uwaterloo.ca/journals/JIS/stevens.html">A Connell-Like Sequence</a>, J. Integer Seqs., 1 (1998), Article 98.1.4.
%H Andrei K. Svinin, <a href="https://arxiv.org/abs/1610.05387">On some class of sums</a>, arXiv:1610.05387 [math.CO], 2016. See p. 7.
%H <a href="/index/Lu#Lyndon">Index entries for sequences related to Lyndon words</a>.
%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F a(n) = (A000217(n+1) - A022003(n-1))/3;
%F a(n) = (A016754(n+1) - A010881(A016754(n+1)))/24;
%F a(n) = (A033996(n+1) - A010881(A033996(n+1)))/24.
%F Euler transform of length 3 sequence [2, 0, 1].
%F a(3*k-1) = k*(3*k + 1)/2;
%F a(3*k) = 3*k*(k + 1)/2;
%F a(3*k+1) = (k + 1)*(3*k + 2)/2.
%F a(n) = floor( (n+1)*(n+2)/6 ) = floor( A000217(n+1)/3 ).
%F a(n+1) = a(n) + A008620(n) = A002264(n+3). - _Reinhard Zumkeller_, Aug 01 2002
%F From _Michael Somos_, Feb 11 2004: (Start)
%F G.f.: x / ((1-x)^2 * (1-x^3)).
%F a(n) = 1 + a(n-1) + a(n-3) - a(n-4).
%F a(-3-n) = a(n). (End)
%F a(n) = a(n-3) + n for n > 2; a(0)=0, a(1)=1, a(2)=2. - _Paul Barry_, Jul 14 2004
%F 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
%F From _Paul Barry_, Apr 16 2005: (Start)
%F 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).
%F a(n) = Sum_{k=0..floor(n/3)} n-3*k. (End)
%F For n > 1, a(n) = A000217(n) - a(n-1) - a(n-2); a(0)=0, a(1)=1.
%F G.f.: x/(1 + x + x^2)/(1 - x)^3. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
%F a(n) = (4 + 3*n^2 + 9*n)/18 + ((n mod 3) - ((n-1) mod 3))/9. - _Klaus Brockhaus_, Oct 01 2009
%F a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5), with n>4, a(0)=0, a(1)=1, a(2)=2, a(3)=3, a(4)=5. - _Harvey P. Dale_, Jul 25 2011
%F a(n) = A214734(n + 2, 1, 3). - _Renzo Benedetti_, Aug 27 2012
%F 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
%F Empirical: a(n) = floor((n+3)/(e^(6/(n+3))-1)). - _Richard R. Forberg_, Jul 24 2013
%F a(n) = Sum_{i=0..n} floor((i+2)/3). - _Bruno Berselli_, Aug 29 2013
%F 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
%F a(n) = n/2 + floor(n^2/3 + 2/3)/2. - _Bruno Berselli_, Jan 23 2017
%F a(n) + a(n+1) = A000212(n+2). - _R. J. Mathar_, Jan 14 2021
%F Sum_{n>=1} 1/a(n) = 20/3 - 2*Pi/sqrt(3). - _Amiram Eldar_, Sep 27 2022
%F E.g.f.: (exp(x)*(4 + 12*x + 3*x^2) - 4*exp(-x/2)*cos(sqrt(3)*x/2))/18. - _Stefano Spezia_, Apr 05 2023
%e 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 + ...
%e 1+2+3=6=t(3), 2+3+5=t(4), 5+7+9=t(5).
%e [n] a(n)
%e --------
%e [1] 1
%e [2] 2
%e [3] 3
%e [4] 1 + 4
%e [5] 2 + 5
%e [6] 3 + 6
%e [7] 1 + 4 + 7
%e [8] 2 + 5 + 8
%e [9] 3 + 6 + 9
%e 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
%p A001840 := n->floor((n+1)*(n+2)/6);
%p A001840:=-1/((z**2+z+1)*(z-1)**3); # conjectured (correctly) by _Simon Plouffe_ in his 1992 dissertation
%p seq(floor(binomial(n-1,2)/3), n=3..61); # _Zerinvary Lajos_, Jan 12 2009
%p 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
%t 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}]
%t f[n_] := Floor[(n + 1)(n + 2)/6]; Array[f, 59, 0] (* Or *)
%t CoefficientList[ Series[ x/((1 + x + x^2)*(1 - x)^3), {x, 0, 58}], x] (* _Robert G. Wilson v_ *)
%t 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 *)
%t LinearRecurrence[{2,-1,1,-2,1},{0,1,2,3,5},60] (* _Harvey P. Dale_, Jul 25 2011 *)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n+4,{3}],#[[1]]<#[[2]]&&#[[1]]<#[[3]]&]],{n,0,15}] (* _Gus Wiseman_, Oct 05 2020 *)
%o (PARI) {a(n) = (n+1) * (n+2) \ 6}; /* _Michael Somos_, Feb 11 2004 */
%o (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
%o (Sage) [binomial(n, 2) // 3 for n in range(2, 61)] # _Zerinvary Lajos_, Dec 01 2009
%o (Haskell)
%o a001840 n = a001840_list !! n
%o a001840_list = scanl (+) 0 a008620_list
%o -- _Reinhard Zumkeller_, Apr 16 2012
%Y Cf. A000031, A001037, A008748, A051168.
%Y Ordered union of triangular matchstick numbers A045943 and generalized pentagonal numbers A001318.
%Y Cf. A058937.
%Y A column of triangle A011847.
%Y Cf. A000217, A130205.
%Y Cf. A258708.
%Y A001399 counts 3-part partitions, ranked by A014612.
%Y A337483 counts either weakly increasing or weakly decreasing triples.
%Y A337484 counts neither strictly increasing nor strictly decreasing triples.
%Y A014311 ranks 3-part compositions, with strict case A337453.
%Y Cf. A007052, A007304, A011782, A069905, A156040, A218004.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
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