Search: id:A000918
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%I A000918 M1599 N0625
%S A000918 1,0,2,6,14,30,62,126,254,510,1022,2046,4094,8190,16382,
%T A000918 32766,65534,131070,262142,524286,1048574,2097150,4194302,
%U A000918 8388606,16777214,33554430,67108862,134217726,268435454
%V A000918 -1,0,2,6,14,30,62,126,254,510,1022,2046,4094,8190,16382,
%W A000918 32766,65534,131070,262142,524286,1048574,2097150,4194302,
%X A000918 8388606,16777214,33554430,67108862,134217726,268435454
%N A000918 2^n - 2.
%C A000918 For n>2, sum(k=1,a(n),(-1)^C(n,k) ) = A064405(a(n))+1 = 0 - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Oct 18 2002
%C A000918 For n > 0, the number of nonempty proper subsets of an n element set. 
               - Ross La Haye (rlahaye(AT)new.rr.com), Feb 07 2004
%C A000918 Numbers n such that abs( sum(k=0,n,(-1)^C(n,k)*C(n+k,n-k)) ) = 1 - Benoit 
               Cloitre (benoit7848c(AT)orange.fr), Jul 03 2004
%C A000918 For n>2 this formula also counts edge rooted forests in a cycle of length 
               n. - Woong Kook (andrewk(AT)math.uri.edu), Sep 08 2004
%C A000918 For n >= 1, conjectured to be the number of integers from 0 to (10^n)-1 
               that lack 0, 1, 2, 3, 4, 5, 6 and 7 as a digit. - Alexandre Wajnberg 
               (alexandre.wajnberg(AT)ulb.ac.be), Apr 25 2005
%C A000918 Beginning with a(2)=2, these are the partial sums of the subsequence 
               of A000079=2^n beginning with A000079(1)=2. Hence for n >= 2 a(n) 
               is the smallest possible sum of exactly one prime, one two-almost 
               prime, one three-almost prime, ... and one (n-1)-almost prime. A060389 
               (partial sums of the primorials, A002110, beginning with A002110(1)=2) 
               is the analogue when all the almost primes must also be squarefree. 
               - Rick L. Shepherd (rshepherd2(AT)hotmail.com), May 20 2005
%C A000918 From the second term on (n>=1), the binary representation of these numbers 
               is a 0 preceded by (n-1) 1's. This pattern (0)111...1110 is the "opposite" 
               of the binary 2^n+1: 1000...0001 (cf. A000051). - Alexandre Wajnberg 
               (alexandre.wajnberg(AT)ulb.ac.be), May 31 2005
%C A000918 The numbers 2^n-2 (n>=2) give the positions of 0's in A110146. Also numbers 
               n such that n^(n+1) = 0 mod (n+2). - Zak Seidov (zakseidov(AT)yahoo.com), 
               Feb 20 2006
%C A000918 Number of surjections from an n-element set onto a two-element set, with 
               n >= 2. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Dec 15 2007
%C A000918 It appears that these are the numbers n such that 3*A135013(n) = n*(n+1), 
               thus answering Problem 2 on the Mathematical Olympiad Foundation 
               of Japan, Final Round Problems, Feb 11 1993.
%C A000918 a(n) = A058896(n)/A052548(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Feb 14 2009]
%C A000918 Let P(A) be the power set of an n-element set A and R be a relation on 
               P(A) such that for all x, y of P(A), xRy if x is a proper subset 
               of y or y is a proper subset of x and x and y are disjoint. Then 
               a(n+1) = |R|. [From Ross La Haye (rlahaye(AT)new.rr.com), Mar 19 
               2009]
%C A000918 a(n) = A164874(n-1,n-1) for n>1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Aug 29 2009]
%C A000918 Apart the first term which is -1 the number of units of a(n) belongs 
               to a periodic sequence: 0, 2, 6, 4. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), 
               Sep 04 2009]
%D A000918 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000918 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000918 H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd 
               ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity 
               Univ., San Antonio, TX, Vol. 2, p. 212.
%D A000918 Mathematical Olympiad Foundation of Japan, Final Round Problems, Feb 
               11 1993, Problem 2.
%D A000918 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 
               33.
%D A000918 A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, 
               Leipzig, 1911, p. 31.
%D A000918 Ross La Haye, Binary Relations on the Power Set of an n-Element Set, 
               Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6. [From 
               Ross La Haye (rlahaye(AT)new.rr.com), Mar 19 2009]
%H A000918 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A000918 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas 
               for Some Functions on Finite Sets</a>
%H A000918 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=77">
               Encyclopedia of Combinatorial Structures 77</a>
%H A000918 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A000918 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000918 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SphereLinePicking.html">Sphere Line Picking</a>
%F A000918 G.f.: 1/(1-2x) - 2/(1-x), e.g.f.: (e^x - 1)^2 - 1. - Dan Fux (dan.fux(AT)OpenGaia.com 
               or danfux(AT)OpenGaia.com), Apr 07 2001
%F A000918 For n>=1, a(n) = A008970(n+1, 2) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Feb 21 2004
%F A000918 G.f.: (3x - 1)/(2x^2 - 3x + 1).
%F A000918 a(n) = 2a(n-1) + 2 - Alexandre Wajnberg (alexandre.wajnberg(AT)ulb.ac.be), 
               Apr 25 2005
%F A000918 a(n) = A000079(n)-2. [From Omar E. Pol (info(AT)polprimos.com), Dec 16 
               2008]
%p A000918 [seq (stirling2(n,2)*2,n=0..28)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Dec 06 2006
%p A000918 ZL := [S, {S=Prod(B,B), B=Set(Z, 1 <= card)}, labeled]: seq(combstruct[count](ZL, 
               size=n), n=0..28); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Mar 13 2007
%p A000918 a:=n->sum (2^j,j=1..n): seq(a(n),n=-1..27); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Oct 03 2007
%p A000918 A000918:=2*z/((2*z-1)*(z-1)); [S. Plouffe in his 1992 dissertation.]
%t A000918 lst={};Do[AppendTo[lst, 2^n-2], {n, 0, 5!}];lst [From Vladimir Orlovsky 
               (4vladimir(AT)gmail.com), Oct 16 2008]
%o A000918 (Other) sage: [gaussian_binomial(n,1,2)-1 for n in xrange(0,29)] # [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 31 2009]
%Y A000918 Row sums of triangle A026998.
%Y A000918 Cf. A000919, A001117, A001118.
%Y A000918 Cf. A095121. A110146.
%Y A000918 Sequence in context: A063452 A009299 A072611 this_sequence A095121 A122958 
               A122959
%Y A000918 Adjacent sequences: A000915 A000916 A000917 this_sequence A000919 A000920 
               A000921
%K A000918 sign,easy
%O A000918 0,3
%A A000918 N. J. A. Sloane (njas(AT)research.att.com).

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