OFFSET
0,5
COMMENTS
Number of subsets with at least 3 elements of an n-element set.
For n>4, number of simple rank-(n-1) matroids over S_n.
Number of non-interval subsets of {1,2,3,...,n} (cf. A000124). - Jose Luis Arregui (arregui(AT)unizar.es), Jun 27 2006
The partial sums of the second diagonal of A008292 or third column of A123125. - Tom Copeland, Sep 09 2008
a(n) is the number of binary sequences of length n having at least three 0's. - Geoffrey Critzer, Feb 11 2009
Starting with "1" = eigensequence of a triangle with the tetrahedral numbers (1, 4, 10, 20, ...) as the left border and the rest 1's. - Gary W. Adamson, Jul 24 2010
a(n) is also the number of crossing set partitions of [n+1] with two blocks. - Peter Luschny, Apr 29 2011
The Kn24 sums, see A180662, of triangle A065941 equal the terms (doubled) of this sequence minus the three leading zeros. - Johannes W. Meijer, Aug 14 2011
From L. Edson Jeffery, Dec 28 2011: (Start)
Nonzero terms of this sequence can be found from the row sums of the fourth sub-triangle extracted from Pascal's triangle as indicated below by braces:
1;
1, 1;
1, 2, 1;
{1}, 3, 3, 1;
{1, 4}, 6, 4, 1;
{1, 5, 10}, 10, 5, 1;
{1, 6, 15, 20}, 15, 6, 1;
... (End)
Partial sums of A000295 (Eulerian Numbers, Column 2).
Second differences equal 2^(n-2) - 1, for n >= 4. - Richard R. Forberg, Jul 11 2013
Starting (0, 0, 1, 5, 16, ...) is the binomial transform of (0, 0, 1, 2, 2, 2, ...). - Gary W. Adamson, Jul 27 2015
a(n - 1) is the rank of the divisor class group of the moduli space of stable rational curves with n marked points, see Keel p. 550. - Harry Richman, Aug 10 2024
REFERENCES
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
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VI: Voronoi Reduction of Three-Dimensional Lattices, Proc. Royal Soc. London, Series A, 436 (1992), 55-68. (See Table 1.)
W. M. B. Dukes, On the number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.
J. Eckhoff, Der Satz von Radon in konvexen Produktstrukturen II, Monat. f. Math., 73 (1969), 7-30.
R. K. Guy, Letter to N. J. A. Sloane
Sean Keel, Intersection theory of moduli space of stable n-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), 545-574.
Pablo Hueso Merino , The first problem from the 55th Spanish Mathematical Olympiad asks to find the value of a(2019) (see comment from Jose Luis Arregui).
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Sergey V. Muravyov, Liudmila I. Khudonogova, and Ekaterina Y. Emelyanova, Interval data fusion with preference aggregation, Measurement (2017), see page 5.
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
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).
FORMULA
G.f.: x^3/((1-2*x)*(1-x)^3).
a(n) = Sum_{k=0..n} binomial(n,k+3) = Sum_{k=3..n} binomial(n,k). - Paul Barry, Jul 30 2004
a(n+1) = 2*a(n) + binomial(n,2). - Paul Barry, Aug 23 2004
(1, 5, 16, 42, 99, ...) = binomial transform of (1, 4, 7, 8, 8, 8, ...). - Gary W. Adamson, Sep 30 2007
E.g.f.: exp(x)*(exp(x)-x^2/2-x-1). - Geoffrey Critzer, Feb 11 2009
a(n) = n - 2 + 3*a(n-1) - 2*a(n-2), for n >= 2. - Richard R. Forberg, Jul 11 2013
For n>1, a(n) = (1/4)*Sum_{k=1..n-2} 2^k*(n-k-1)*(n-k). For example, (1/4)*(2^1*(4*5) + 2^2*(3*4) + 2^3*(2*3) + 2^4*(1*2)) = 168/4 = 42. - J. M. Bergot, May 27 2014 [edited by Danny Rorabaugh, Apr 19 2015]
a(n) = Sum_{k=1..n-2} Sum_{i=1..n} (n-k-1) * C(k,i). - Wesley Ivan Hurt, Sep 19 2017
a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4) for n > 3. - Chai Wah Wu, Apr 03 2021
a(n) = a(n-1) + 1 + A000247(n-1). - Harry Richman, Aug 13 2024
EXAMPLE
a(4) = 5 is the number of crossing set partitions of {1,2,..,5}, card{13|245, 14|235, 24|135, 25|134, 35|124}. - Peter Luschny, Apr 29 2011
MAPLE
A002662 := z**2/(2*z-1)/(z-1)**3; # conjectured by Simon Plouffe in his 1992 dissertation
A002662 := proc(n): 2^n - 1 - n*(n+1)/2 end: seq(A002662(n), n=0..33); # Johannes W. Meijer, Aug 14 2011
MATHEMATICA
With[{nn=40}, Join[{0}, First[#]-1-Last[#]&/@Thread[{2^Range[nn], Accumulate[ Range[nn]]}]]] (* Harvey P. Dale, May 10 2012 *)
Table[2^n - Binomial[n, 2] - n - 1, {n, 1, 100}] (* Pablo Hueso Merino, Dec 17 2019 *)
PROG
(Magma) [2^n - 1 - n*(n+1)/2: n in [0..35]]; // Vincenzo Librandi, May 20 2011
(Haskell)
a002662 n = a002662_list !! n
a002662_list = map (sum . drop 3) a007318_tabl
-- Reinhard Zumkeller, Jun 20 2015
(PARI) a(n)=2^n-1-n*(n+1)/2 \\ Charles R Greathouse IV, Oct 11 2015
(Python)
def A002662(n): return (1<<n)-1-(n*(n+1)>>1) # Chai Wah Wu, Aug 29 2023
CROSSREFS
KEYWORD
easy,nonn,nice
AUTHOR
STATUS
approved