A001840 Expansion of g.f. x/((1 - x)^2*(1 - x^3)).
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
Offset: 0
Examples
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
References
- Tom 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.
- 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).
- 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).
- 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
- Chwas Ahmed, Paul Martin, and Volodymyr Mazorchuk, On the number of principal ideals in d-tonal partition monoids, arXiv preprint arXiv:1503.06718 [math.CO], 2015.
- Gert Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
- David 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.
- David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See p. 19.
- Cristian Cobeli, Aaditya Raghavan, and Alexandru Zaharescu, On the central ball in a translation invariant involutive field, arXiv:2408.01864 [math.NT], 2024. See p. 7.
- Neville de Mestre and John Baker, Pebbles, Ducks and Other Surprises, Australian Maths. Teacher, Vol. 48, No 3, 1992, pp. 4-7.
- Peter M. Chema, Illustration of first 27 terms as corners of a double hexagon spiral from 0
- 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]
- Richard 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, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
- Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
- R. P. Loh, A. G. Shannon, and 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; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
- Gerhard Post and G. J. Woeginger, Sports tournaments, home-away assignments and the break minimization problem, Discrete Optimization 3, pp. 165-173, 2006.
- Michael Somos, Somos Polynomials.
- Gary E. Stevens, A Connell-Like Sequence, J. Integer Seqs., 1 (1998), Article 98.1.4.
- Andrei K. Svinin, On some class of sums, arXiv:1610.05387 [math.CO], 2016. See p. 7.
- Eric Weisstein's World of Mathematics, Lower Matching Number.
- Eric Weisstein's World of Mathematics, Triangular Grid Graph.
- Eric Weisstein's World of Mathematics, Triangular Honeycomb King Graph.
- 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).
Crossrefs
Programs
-
Haskell
a001840 n = a001840_list !! n a001840_list = scanl (+) 0 a008620_list -- Reinhard Zumkeller, Apr 16 2012
-
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
-
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 *) Table[Length[Select[Join@@Permutations/@IntegerPartitions[n+4,{3}],#[[1]]<#[[2]]&&#[[1]]<#[[3]]&]],{n,0,15}] (* Gus Wiseman, Oct 05 2020 *) -
PARI
{a(n) = (n+1) * (n+2) \ 6}; /* Michael Somos, Feb 11 2004 */ -
Sage
[binomial(n, 2) // 3 for n in range(2, 61)] # Zerinvary Lajos, Dec 01 2009
Formula
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 ).
From Michael Somos, Feb 11 2004: (Start)
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). (End)
a(n) = a(n-3) + n for n > 2; 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
From Paul Barry, Apr 16 2005: (Start)
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. (End)
For n > 1, a(n) = A000217(n) - a(n-1) - a(n-2); a(0)=0, a(1)=1.
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 n>4, 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_{i=0..n} floor((i+2)/3). - 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
a(n) = n/2 + floor(n^2/3 + 2/3)/2. - Bruno Berselli, Jan 23 2017
a(n) + a(n+1) = A000212(n+2). - R. J. Mathar, Jan 14 2021
Sum_{n>=1} 1/a(n) = 20/3 - 2*Pi/sqrt(3). - Amiram Eldar, Sep 27 2022
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
Comments