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A002662
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2^n - 1 - n*(n+1)/2.
(Formerly M3866 N1585)
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16
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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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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. [From Tom Copeland, Sep 09 2008]
a(n) is the number of binary sequences of length n having at least three 0's. [From 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]
Contribution 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)
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REFERENCES
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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.)
J. Eckhoff, Der Satz von Radon in konvexen Productstrukturen II, Monat. f. Math., 73 (1969), 7-30.
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
_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.
W. M. B. Dukes, On the number of matroids on a finite set
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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, C(n,k+3)} = sum{k=3..n, C(n,k)}. - Paul Barry, Jul 30 2004
a(n) = 2*a(n-1)+C(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). [From Geoffrey Critzer, Feb 11 2009]
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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 [From Vladimir Orlovsky, Jan 10 2009]
With[{nn=40}, Join[{0}, First[#]-1-Last[#]&/@Thread[{2^Range[nn], Accumulate[ Range[nn]]}]]] (* From 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
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CROSSREFS
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a(n)= A055248(n,3). Partial sums of A000295.
Cf. A000079, A000225, A000295, A002663, A002664, A035038-A035042.
Sequence in context: A097810 A187004 A055796 * A143962 A066634 A034358
Adjacent sequences: A002659 A002660 A002661 * A002663 A002664 A002665
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KEYWORD
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easy,nonn,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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