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A002662
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a(n) = 2^n - 1 - n*(n+1)/2.
(Formerly M3866 N1585)
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18
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0, 0, 0, 1, 5, 16, 42, 99, 219, 466, 968, 1981, 4017, 8100, 16278, 32647, 65399, 130918, 261972, 524097, 1048365, 2096920, 4194050, 8388331, 16776915, 33554106, 67108512, 134217349, 268435049, 536870476, 1073741358, 2147483151, 4294966767, 8589934030
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OFFSET
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0,5
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COMMENTS
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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
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REFERENCES
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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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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.
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.
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FORMULA
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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_{0<k<n-1} 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]
Convolution of A001045 and (A000290 shifted by one place). - Oboifeng Dira, Aug 16 2016
a(n) = Sum_{k=1..n-2} Sum_{i=1..n} (n-k-1) * C(k,i). - Wesley Ivan Hurt, Sep 19 2017
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EXAMPLE
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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
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MAPLE
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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
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MATHEMATICA
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a=1; lst={}; s1=s2=s3=0; Do[s1+=a; s2+=s1; s3+=s2; AppendTo[lst, s3]; a=a*2, {n, 6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 10 2009 *)
With[{nn=40}, Join[{0}, First[#]-1-Last[#]&/@Thread[{2^Range[nn], Accumulate[ Range[nn]]}]]] (* Harvey P. Dale, May 10 2012 *)
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PROG
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(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
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CROSSREFS
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a(n)= A055248(n,3).
Cf. A000079, A000225, A000295, A002663, A002664, A035038-A035042, A007318.
Cf. also A000290, A001045.
Sequence in context: A187004 A255135 A055796 * A143962 A066634 A241794
Adjacent sequences: A002659 A002660 A002661 * A002663 A002664 A002665
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KEYWORD
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easy,nonn,nice,changed
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AUTHOR
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N. J. A. Sloane, Apr 30 1991
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STATUS
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approved
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