0,2

Number of subsets of an n-set.

There are 2^(n-1) compositions (ordered partitions) of n - see for example Riordan. This is the unlabeled analogue of the preferential labelings sequence A000670.

This is also the number of weakly unimodal permutations of 1..n, that is, permutations with exactly one local maximum. E.g. a(5)=16: 12345, 12354, 12453, 12543, 13452, 13542, 14532 and 15432 and their reversals. - Jon Perry (perry(AT)globalnet.co.uk), Jul 27 2003. Proof: see next line! See also A087783.

Proof: n must appear somewhere and there are 2^(n-1) possible choices for the subset that precedes it. These must appear in increasing order and the rest must follow n in decreasing order. QED. - N. J. A. Sloane (njas(AT)research.att.com), Oct 26, 2003.

a(n+1) = smallest number that is not the sum of any number of (distinct) earlier terms.

Same as Pisot sequences E(1,2), L(1,2), P(1,2), T(1,2). See A008776 for definitions of Pisot sequences.

With initial 1 omitted, same as Pisot sequences E(2,4), L(2,4), P(2,4), T(2,4). - David W. Wilson.

Not the sum of two or more consecutive numbers. - Lekraj Beedassy (blekraj(AT)yahoo.com), May 14 2004

Least deficient or near-perfect numbers (i.e. n such that sigma(n)=A000203(n)=2n-1). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 03 2004. Comment from Max Alekseyev (maxale(AT)gmail.com), Jan 26 2005: All the powers of 2 are least deficient numbers but it is not known if there exists a least deficient number not a power of 2.

The sum of the numbers in the n-th row of Pascal's triangle; the sum of the coefficients of x in the expansion of (x+1)^n.

The only hailstone sequence which doesn't rebound (except "on the ground"). - Alexandre Wajnberg (alexandre.wajnberg(AT)ulb.ac.be), Jan 29 2005

With p(n) = the number of integer partitions of n, p(i) = the number of parts of the i-th partition of n, d(i) = the number of different parts of the i-th partition of n, m(i,j) = multiplicity of the j-th part of the i-th partition of n, sum_{i=1}^{p(n)} = sum over i and prod_{j=1}^{d(i)} = product over j one has: a(n)=sum_{i=1}^{p(n)} p(i)!/(prod_{j=1}^{d(i)} m(i,j)!) - Thomas Wieder (wieder.thomas(AT)t-online.de), May 18 2005

a(n+1) = a(n) XOR 3a(n) where XOR is binary exclusive OR operator. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 19 2005

The number of binary relations on an n-element set that are both symmetric and antisymmetric. Also the number of binary relations on an n-element set that are symmetric, antisymmetric and transitive.

An autocopy sequence: its first differences are the sequence itself. - Alexandre Wajnberg & Eric Angelini (alexandre.wajnberg(AT)ulb.ac.be), Sep 07 2005

a(n) = largest number with shortest addition chain involving n additions. - David W. Wilson (davidwwilson(AT)comcast.net), Apr 23 2006

Beginning with a(1) = 0, numbers not equal to the sum of previous distinct natural numbers. - Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Aug 06 2006

Smallest order of exactly p(n) nonisomorphic Abelian groups, where p(n)=A000041(n). {First occurrence of p(n) in A000688(n)} - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 11 2006

For n>=1, a(n) is equal to the number of functions f:{1,2...,n}->{1,2} such that for a fixed x in {1,2,...,n} and a fixed y in {1,2]} we have f(x)<>y. - Aleksandar M. Janjic and Milan R. Janjic (agnus(AT)blic.net), Mar 27 2007

Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which x = y. - Ross La Haye (rlahaye(AT)new.rr.com), Jan 09 2008 Ross La Haye

a(n)= the number of different ways to run up a staircase with n steps, taking steps of sizes 1,2,3,... and r (r<=n), where the order IS important and there is no restriction on the number or the size of each step taken. - Mohammad K. Azarian (azarian(AT)evansville.edu), May 21 2008

a(n)=number of permutations on [n+1] such that every initial segment is an interval of integers. Example: a(3) counts 1234, 2134, 2314, 2341, 3214, 3241, 3421, 4321. The map "p -> ascents of p" is a bijection from these permutations to subsets of [n]. An ascent of a permutation p is a position i such that p(i) < p(i+1). The permutations shown map to 123, 23, 13, 12, 3, 2, 1 and the empty set respectively. - David Callan (callan(AT)stat.wisc.edu), Jul 25 2008

2^(n-1) is the largest number having n divisors (in the sense of A077569); A005179(n) is the smallest. [From T. D. Noe (noe(AT)sspectra.com), Sep 02 2008]

Contribution from Bill R McEachen (bmceachen(AT)centralsan.org), Oct 29 2008: (Start)

a(n) appears to match the number of divisors of the modified primorials (excluding 2,3and 5)

Very limited range examined, PARI example shown (End)

Successive k such that EulerPhi[k]/k = 1/2. [From Artur Jasinski (grafix(AT)csl.pl), Nov 07 2008]

A classical transform consists (for general a(n)) in swapping a(2n) and a(2n+1);examples for Jacobsthal A001045 and successive differences: A092808,A094359,A140505. a(n)=A000079 leads to 2,1,8,4,32,16,=A135520. [From Paul Curtz (bpcrtz(AT)free.fr), Jan 05 2009]

This is also the (L)-sieve transform of {2,4,6,8,...,2n,...}=A005843. (See A152079 for the definition of the (L)-sieve transform.) [From John W. Layman (layman(AT)math.vt.edu), Jan 23 2009]

a(n) = a(n-1)-th even natural numbers (A005843) for n > 1. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Apr 25 2009]

For n >= 0, a(n) is the number of leaves in a complete binary tree of height n. For n > 0, a(n) is the number of nodes in an n-cube. [From Kailasam Viswanathan Iyer (kvi(AT)nitt.edu), May 04 2009]

Permutations of n+1 elements where no element is more than one position left of its original place. For example, there are 4 such permutations of three elements: 123, 132, 213, and 312. The 8 such permutations of four elements are 1234, 1243, 1324, 1423, 2134, 2143, 3124, and 4123. [From Joerg Arndt (arndt(AT)jjj.de), June 24 2009]

Catalan transform of A099087. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 29 2009]

a(n) written in base 2: 1,10,100,1000,10000,..., i.e. (n+1)times 1, n times 0 (A011557(n)). [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Aug 02 2009]

Except for the first term, number n such that if A=(7/8)*n^4; B=(7/16)*n^4; C=(17/16)*n^4; D=(5/4)*n^4; then A^3+B^3+C^3=D^3 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Sep 08 2009]

Or, phi(n) is equal to the number of perfect partitions of n. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 10 2009]

These are the 2-smooth numbers, positive integers with no prime factors greater than 2. [From Michael Porter (michael_b_porter(AT)yahoo.com), Oct 04 2009]

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 1016.

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education Journal, Vol. 31, No. 1, pp. 24-28, Winter 1997.

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

R. Ondrejka, Exact values of 2^n, n=1(1)4000, Math. Comp., 23 (1969), 456.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 124.

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).

Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

N. J. A. Sloane, Table of n, 2^n for n = 0..1000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Henry Bottomley, Illustration of initial terms

D. Butler, Powers of Two up to 2^222

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 18

Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors ..., J. Integer Seqs., Vol. 6, 2003.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 6

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 68

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 72

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 267

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

G. Villemin's Almanac of Numbers, Puissances de 2

Sage Weil, 1058 powers of two

Eric Weisstein's World of Mathematics, Fractional Part

Eric Weisstein's World of Mathematics, PowerFractional Parts

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (1).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (2).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (3).

Eric Weisstein's World of Mathematics, Hypercube

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics(4)

Eric Weisstein's World of Mathematics, Hailstone Number

Eric Weisstein's World of Mathematics, Erf

Eric Weisstein's World of Mathematics, Abundance

Wikipedia, Almost perfect number

Index entries for "core" sequences

Index entries for related partition-counting sequences

Index entries for sequences related to linear recurrences with constant coefficients

a(n) = 2^n; a(n) = 2*a(n-1). G.f.: 1/(1-2x), e.g.f.: exp(2x).

2^n = Sum_{k=0..n} binomial(n, k).

a(n) is the number of occurrences of n in A000523. a(n) = A001045(n) + A001045(n+1). a(n) = 1 + sum_{k=0..(n-1)} a(k). The Hankel transform of this sequence gives A000007 = [1, 0, 0, 0, 0, 0, ...]. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 25 2004

n such that phi(n)=n/2, for n>1, where phi is the Euler's totient (A000010). - Lekraj Beedassy (lbeedassy(AT)hotmail.com), Sep 07 2004

This sequence can be generated by the following formula: a(n) = a(n-1) + 2*a(n-2) when n > 2; a[1] = 1, a[2] = 2 - Alex Vinokur (alexvn(AT)barak-online.net), Oct 24 2004

a(n) = StirlingS2(n+1,2) + 1 - Ross La Haye (rlahaye(AT)new.rr.com), Jan 09 2008 Ross La Haye

This sequence can be generated by a(n+2)=6a(n+1)-8a(n), n=1,2,3,... with a(1)=1, a(2)=2. - Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Aug 06 2008

a(n)=ka(n-1)+(4-2k)a(n-2) for any integer k and n>1, with a(0)=1, a(1)=2. [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Dec 05 2008]

Equals the partition numbers A000041 convolved with A152537. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 06 2008]

Formula from Thomas Wieder (wieder.thomas(AT)t-online.de), Feb 25 2009:

a(n) = sum_{l_1=0}^{n+1} sum_{l_2=0}^{n}...sum_{l_i=0}^{n-i}...sum_{l_n=0}^{1}

delta(l_1,l_2,...,l_i,...,l_n)

where delta(l_1,l_2,...,l_i,...,l_n) = 0 if any l_i <= l_(i+1) and l_(i+1) <> 0

and delta(l_1,l_2,...,l_i,...,l_n) = 1 otherwise.

G.f.: exp(x)*cosh(x) . [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 05 2009]

a(0)=1, a(1)=2; a(n)=a(n-1)^2/a(n-2), n>=2 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Sep 22 2009]

A000010(a(n))=A002033(a(n)). [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 10 2009]

There are 2^3 = 8 subsets of a 3-element set {1,2,3}, namely { -, 1, 2, 3, 12, 13, 23, 123 }.

For n=2, A=14, B=7, C=17, D=20, and 14^3+7^3+17^3=20^3 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jun 25 2009]

A000079 := n->2^n; [ seq(2^n, n=0..50) ];

with(combstruct); SeqSetU := [S, {S=Sequence(U), U=Set(Z, card >= 1)}, unlabeled]; seq(count(SeqSetU, size=j), j=1..12);

with(combinat):seq(stirling2(n, 2)+1, n=1..34); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 04 2007

seq(binomial(n+0, 0)*2^n, n=0..33); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 08 2008

with(finance):seq(futurevalue(2, 1, n), n=-1..31); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 24 2009]

restart: G(x):=exp(x)*cosh(x): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=1..34 ); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 05 2009]

Array[ 2^#&, 50, 0 ]

a = {}; Do[If[EulerPhi[x]/x == 1/2, AppendTo[a, x]], {x, 1, 2048}]; a [From Artur Jasinski (grafix(AT)csl.pl), Nov 07 2008]

(PARI) a(n)=if(n<0, 0, 2^n)

(PARI) { unimodal(n)=local(x, d, um, umc); umc=0; for (c=0, n!-1, x=numtoperm(n, c); d=0; um=1; for (j=2, n, if (x[j]<x[j-1], d=1); if (x[j]>x[j-1] && d==1, um=0); if (um==0, break)); if (um==1, print(x)); umc+=um); umc }

sage: [lucas_number2(n, 4, 4) for n in xrange(-1, 27)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 08 2008

(PARI) a=7*11*13*17*19*23*29*31*37*41*43*47*53*59*61 %32 = 3909612711980232366109 ? b=numdiv(a) %33 = 32768 [From Bill R McEachen (bmceachen(AT)centralsan.org), Oct 29 2008]

(PARI) { x=1; for (n=0, 1000, write("b000079.txt", n, " ", x); x+=x); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 26 2009]

a(n) = 2*A001045(n)+A078008(n) = 3*A001045(n)+(-1)^n. - Paul Barry (pbarry(AT)wit.ie), Feb 20 2003

Cf. A000225.

A000079 is the Hankel transform (see A001906 for the definition) of A000984, A002426, A026375, A026387, A026569, A026585, A026671 and A032351 - John W. Layman (layman(AT)math.vt.edu), Jul 31 2000

Euler transform of A001037.

Complement of A057716.

a(n) = A118654(n, 2).

a(n) = A140740(n+1, 1).

Cf. A038754, A133464, A140730, A037124.

Cf. A001787, A001788, A001789, A003472, A054849, A002409, A054851, A140325, A140354.

Cf. A000041, A152537 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 06 2008]

Equals row sums of the partition convolution triangle, A152538 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 10 2008]

Cf. A000010, A002033.[From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 10 2009]

Sequence in context: A166444 A084633 A122803 this_sequence A120617 A050732 A138815

Adjacent sequences: A000076 A000077 A000078 this_sequence A000080 A000081 A000082

core,easy,nice,nonn

N. J. A. Sloane (njas(AT)research.att.com).

Clarified a comment T. D. Noe (noe(AT)sspectra.com), Aug 30 2009