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A000918
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2^n - 2.
(Formerly M1599 N0625)
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60
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-1, 0, 2, 6, 14, 30, 62, 126, 254, 510, 1022, 2046, 4094, 8190, 16382, 32766, 65534, 131070, 262142, 524286, 1048574, 2097150, 4194302, 8388606, 16777214, 33554430, 67108862, 134217726, 268435454
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| For n>3, a(n) is the expected number of tosses of a fair coin to get (n-1) consecutive heads - Pratik Poddar, Feb 04 2011
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
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
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
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
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
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
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
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
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
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.
a(n) = A058896(n)/A052548(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 14 2009]
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]
a(n) = A164874(n-1,n-1) for n>1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 29 2009]
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]
The permutahedron Pi_n has 2^n-2 facets [Pashkovich]. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 17 2009]
a(n) = A173787(n,1); a(n) = A028399(2*n)/A052548(n), n>0. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 28 2010]
First differences of A005803. [Reinhard Zumkeller, Oct 12 2011]
For n>=1, a(n+1) is the smallest even number with bit sum n. Cf. A069532. - Jason Kimberley, Nov 01 2011
a(n) is the number of branches of a complete binary tree of n levels. - Denis Lorrain, Dec 16 2011
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REFERENCES
| 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.
Ralph P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, Addison-Wesley, 2004, p. 134. [From Mohammad K. Azarian, October 27 2011.]
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]
Mathematical Olympiad Foundation of Japan, Final Round Problems, Feb 11 1993, Problem 2.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
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).
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. A. HILL, M. J. HOPKINS AND D. C. RAVENEL, ON THE NON-EXISTENCE OF ELEMENTS OF KERVAIRE INVARIANT ONE [From N. J. A. Sloane, Sep 27 2010]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 77
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Kanstantsin Pashkovich, Symmetry in Extended Formulations of the Permutahedron, Dec 17, 2009. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 17 2009]
S. 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.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Sphere Line Picking
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-2)
Pratik Poddar, Consecutive Heads Puzzle, Dec 2009
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FORMULA
| 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
For n>=1, a(n) = A008970(n+1, 2) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 21 2004
G.f.: (3x-1)/((2x-1)(x-1)).
a(n) = 2a(n-1) + 2 - Alexandre Wajnberg (alexandre.wajnberg(AT)ulb.ac.be), Apr 25 2005
a(n) = A000079(n)-2. [From Omar E. Pol (info(AT)polprimos.com), Dec 16 2008]
a(n+1) = A027383(2n-1). - Jason Kimberley, Nov 02 2011
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MAPLE
| [seq (stirling2(n, 2)*2, n=0..28)]; # Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 06 2006
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
a:=n->sum (2^j, j=1..n): seq(a(n), n=-1..27); # Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 03 2007
A000918:=2*z/((2*z-1)*(z-1)); # S. Plouffe in his 1992 dissertation for the sequence without the leading -1.
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MATHEMATICA
| lst={}; Do[AppendTo[lst, 2^n-2], {n, 0, 5!}]; lst (* From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 16 2008 *)
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PROG
| (Other) sage: [gaussian_binomial(n, 1, 2)-1 for n in xrange(0, 29)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 31 2009]
(PARI) a(n)=2^n-2 \\ Charles R Greathouse IV, Jun 16 2011
(MAGMA) [2^n - 2: n in [0..40]]; // Vincenzo Librandi, Jun 23 2011
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CROSSREFS
| Row sums of triangle A026998.
Cf. A000919, A001117, A001118, A095121. A110146.
Sequence in context: A072611 A192966 * A122958 A095121 A122959 A059076
Adjacent sequences: A000915 A000916 A000917 * A000919 A000920 A000921
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KEYWORD
| sign,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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