login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000079 Powers of 2: a(n) = 2^n.
(Formerly M1129 N0432)
1905

%I M1129 N0432

%S 1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536,

%T 131072,262144,524288,1048576,2097152,4194304,8388608,16777216,

%U 33554432,67108864,134217728,268435456,536870912,1073741824,2147483648,4294967296,8589934592

%N Powers of 2: a(n) = 2^n.

%C 2^0 = 1 is the only odd power of 2.

%C Number of subsets of an n-set.

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

%C This is also the number of weakly unimodal permutations of 1..n+1, that is, permutations with exactly one local maximum. E.g., a(4)=16: 12345, 12354, 12453, 12543, 13452, 13542, 14532 and 15432 and their reversals. - _Jon Perry_, Jul 27 2003. Proof: see next line! See also A087783.

%C 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_, Oct 26 2003

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

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

%C With initial 1 omitted, same as Pisot sequences E(2,4), L(2,4), P(2,4), T(2,4). - _David W. Wilson_

%C Not the sum of two or more consecutive numbers. - _Lekraj Beedassy_, May 14 2004

%C Least deficient or near-perfect numbers (i.e., n such that sigma(n)=A000203(n)=2n-1). - _Lekraj Beedassy_, Jun 03 2004. [Comment from _Max Alekseyev_, Jan 26 2005: All the powers of 2 are least deficient numbers but it is not known if there exists a least deficient number that is not a power of 2.]

%C Almost-perfect numbers referred to as least deficient or slightly defective (Singh 1997) numbers. Does "near-perfect numbers" refer to both almost-perfect numbers (sigma(n) = 2n-1) and quasi-perfect numbers (sigma(n) = 2n+1)? There are no known quasi-perfect or least abundant or slightly excessive (Singh 1997) numbers.

%C 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.

%C The Collatz conjecture (the hailstone sequence will eventually reach the number 1, regardless of which positive integer is chosen initially) may be restated as (the hailstone sequence will eventually reach a power of 2, regardless of which positive integer is chosen initially).

%C The only hailstone sequence which doesn't rebound (except "on the ground"). - _Alexandre Wajnberg_, Jan 29 2005

%C 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, one has: a(n) = sum_{i=1..p(n)} (p(i)! / (prod_{j=1..d(i)} m(i,j)!)). - _Thomas Wieder_, May 18 2005

%C a(n+1) = a(n) XOR 3a(n) where XOR is the binary exclusive OR operator. - _Philippe Deléham_, Jun 19 2005

%C 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.

%C The first differences are the sequence itself. - _Alexandre Wajnberg_ and _Eric Angelini_, Sep 07 2005

%C a(n) = largest number with shortest addition chain involving n additions. - _David W. Wilson_, Apr 23 2006

%C Beginning with a(1) = 0, numbers not equal to the sum of previous distinct natural numbers. - _Giovanni Teofilatto_, Aug 06 2006

%C 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 Janjic_, Mar 27 2007

%C 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_, Jan 09 2008

%C 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_, May 21 2008

%C 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_, Jul 25 2008

%C 2^(n-1) is the largest number having n divisors (in the sense of A077569); A005179(n) is the smallest. - _T. D. Noe_, Sep 02 2008

%C a(n) appears to match the number of divisors of the modified primorials (excluding 2, 3 and 5). Very limited range examined, PARI example shown. - _Bill McEachen_, Oct 29 2008

%C Successive k such that EulerPhi[k]/k = 1/2. - _Artur Jasinski_, Nov 07 2008

%C 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. - _Paul Curtz_, Jan 05 2009

%C This is also the (L)-sieve transform of {2,4,6,8,...,2n,...} = A005843. (See A152009 for the definition of the (L)-sieve transform.) - _John W. Layman_, Jan 23 2009

%C a(n) = a(n-1)-th even natural numbers (A005843) for n > 1. - _Jaroslav Krizek_, Apr 25 2009

%C 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. - _K.V.Iyer_, May 04 2009

%C Permutations of n+1 elements where no element is more than one position right 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. - _Joerg Arndt_, Jun 24 2009

%C Catalan transform of A099087. - _R. J. Mathar_, Jun 29 2009

%C a(n) written in base 2: 1,10,100,1000,10000,..., i.e., (n+1) times 1, n times 0 (A011557(n)). - _Jaroslav Krizek_, Aug 02 2009

%C Or, phi(n) is equal to the number of perfect partitions of n. - _Juri-Stepan Gerasimov_, Oct 10 2009

%C These are the 2-smooth numbers, positive integers with no prime factors greater than 2. - _Michael B. Porter_, Oct 04 2009

%C A064614(a(n)) = A000244(n) and A064614(m) < A000244(n) for m<a(n). - _Reinhard Zumkeller_, Feb 08 2010

%C a(n) = the largest number m such that the number of steps of iterations of {r - (largest divisor d < r)} needed to reach 1 starting at r = m is equal to n. Example (a(5) = 32): 32 - 16 = 16; 16 - 8 = 8; 8 - 4 = 4; 4 - 2 = 2; 2 - 1 = 1; number 32 has 5 steps and is the largest such number. See A105017, A064097, A175125. - _Jaroslav Krizek_, Feb 15 2010

%C a(n) = A173786(n,n)/2 = A173787(n+1,n). - _Reinhard Zumkeller_, Feb 28 2010

%C a(n) is the smallest multiple of a(n-1). - _Dominick Cancilla_, Aug 09 2010

%C The powers-of-2 triangle T(n,k), n>=0 and 0<=k<=n, begins with: {1}; {2, 4}; {8, 16, 32}; {64, 128, 256, 512}; ... . The first left hand diagonal T(n,0) = A006125(n+1), the first right hand diagonal T(n,n) = A036442(n+1) and the center diagonal T(2*n,n) = A053765(n+1). Some triangle sums, see A180662, are: Row1(n) = A122743(n), Row2(n) = A181174(n), Fi1(n) = A181175(n), Fi2(2*n) = A181175(2*n) and Fi2(2*n+1) = 2*A181175(2*n+1). - _Johannes W. Meijer_, Oct 10 2010

%C Records in the number of prime factors. - _Juri-Stepan Gerasimov_, Mar 12 2011

%C Row sums of A152538. - _Gary W. Adamson_, Dec 10 2008

%C A078719(a(n)) = 1; A006667(a(n)) = 0. - _Reinhard Zumkeller_, Oct 08 2011

%C The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=1, a(n) equals the number of 2-colored compositions of n such that no adjacent parts have the same color. - _Milan Janjic_, Nov 17 2011

%C Equals A001405 convolved with its right-shifted variant: (1 + 2x + 4x^2 + ...) = (1 + x + 2x^2 + 3x^3 + 6x^4 + 10x^5 + ...) * (1 + x + x^2 + 2x^3 + 3x^4 + 6x^5 + ...). - _Gary W. Adamson_, Nov 23 2011

%C The number of odd-sized subsets of an n+1-set. For example, there are 2^3 odd-sized subsets of {1,2,3,4}, namely {1}, {2}, {3}, {4}, {1,2,3}, {1,2,4}, {1,3,4}, and {2,3,4}. Also, note that 2^n=sum(C(n+1,2k-1),k=1..floor(n/2+1/2)). - _Dennis P. Walsh_, Dec 15 2011

%C a(n) = number of 1's in any row of Pascal's triangle (mod 2) whose row number has exactly n 1's in its binary expansion (see A007318 and A047999). (The result of putting together A001316 and A000120.) - _Marcus Jaiclin_, Jan 31 2012

%C A204455(k) = 1 if and only if k is in this sequence. - _Wolfdieter Lang_, Feb 04 2012

%C A209229(a(n)) = 1. - _Reinhard Zumkeller_, Mar 07 2012

%C A001227(a(n)) = 1. - _Reinhard Zumkeller_, May 01 2012

%C For n>=1 apparently the number of distinct finite languages over a unary alphabet, whose minimum regular expression has alphabetic width n (verified up to n=17), see the Gruber/Lee/Shallit link. - _Hermann Gruber_, May 09 2012

%C First differences of A000225. - _Omar E. Pol_, Feb 19 2013

%C This is the lexicographically earliest sequence which contains no arithmetic progression of length 3. - Daniel E. Frohardt, Apr 03 2013

%C a(n-2) is the number of bipartitions of {1..n} (i.e., set partitions into two parts) such that 1 and 2 are not in the same subset. - _Jon Perry_, May 19 2013

%C Numbers n such that the n-th cyclotomic polynomial has a root mod 2; numbers n such that the n-th cyclotomic polynomial has an even number of odd coefficients. - _Eric M. Schmidt_, Jul 31 2013

%C More is known now about non-power-of-2 "Almost Perfect Numbers" as described in Dagal. - _Jonathan Vos Post_, Sep 01 2013

%C Number of symmetric Ferrers diagrams that fit into an n X n box. - _Graham H. Hawkes_, Oct 18 2013

%C Numbers n such that sigma(2n) = 2n + sigma(n). - _Jahangeer Kholdi_, Nov 23 2013

%C a(1), ..., a(floor(n/2)) are all values of permanent on set of square (0,1)-matrices of order n>=2 with row and column sums 2. - _Vladimir Shevelev_, Nov 26 2013

%C Numbers whose base-2 expansion has exactly one bit set to 1, and thus has base-2 sum of digits equal to one. - _Stanislav Sykora_, Nov 29 2013

%C A072219(a(n)) = 1. - _Reinhard Zumkeller_, Feb 20 2014

%C a(n) is the largest number k such that (k^n-2)/(k-2) is an integer (for n > 1); (k^a(n)+1)/(k+1) is never an integer (for k > 1 and n > 0). - _Derek Orr_, May 22 2014

%C If x = A083420(n), y = a(n+1) and z = A087289(n), then x^2+2*y^2 = z^2. - _Vincenzo Librandi_, Jun 09 2014

%C The mini-sequence b(n) = least number k > 0 such that 2^k ends in n identical digits is given by {1, 18, 39}. The repeating digits are {2, 4, 8} respectively. Note that these are consecutive powers of 2 (2^1, 2^2, 2^3), and these are the only powers of 2 (2^k, k > 0) that are only one digit. Further, this sequence is finite. The number of n-digit endings for a power of 2 with n or more digits id 4*5^(n-1). Thus, for b(4) to exist, one only needs to check exponents up to 4*5^3 = 500. Since b(4) does not exist, it is clear that no other number will exist. - _Derek Orr_, Jun 14 2014

%C The least number k > 0 such that 2^k ends in n consecutive decreasing digits is a 3-number sequence given by {1, 5, 25}. The consecutive decreasing digits are {2, 32, 432}. There are 100 different 3-digit endings for 2^k. There are no k-values such that 2^k ends in '987', '876', '765', '654', '543', '321', or '210'. The k-values for which 2^k ends in '432' are given by 25 mod 100. For k = 25 + 100*x, the digit immediately before the run of '432' is {4, 6, 8, 0, 2, 4, 6, 8, 0, 2, ...} for x = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...}, respectively. Thus, we see the digit before '432' will never be a 5. So, this sequence is complete. - _Derek Orr_, Jul 03 2014

%C a(n) is the number of permutations of length n avoiding both 231 and 321 in the classical sense which are breadth-first search reading words of increasing unary-binary trees. For more details, see the entry for permutations avoiding 231 at A245898. - _Manda Riehl_, Aug 05 2014

%C Numbers n such that sigma(n) = sigma(2n) - phi(4n). - _Farideh Firoozbakht_, Aug 14 2014

%C This is a B_2 sequence: for i < j, differences a(j) - a(i) are all distinct. Here 2*a(n) < a(n+1) + 1, so a(n) - a(0) < a(n+1) - a(n). - _Thomas Ordowski_, Sep 23 2014

%C a(n) counts n-walks (closed) on the graph G(1-vertex; 1-loop, 1-loop). - _David Neil McGrath_, Dec 11 2014

%C a(n-1) counts walks (closed) on the graph G(1-vertex; 1-loop, 2-loop, 3-loop, 4-loop, ...). - _David Neil McGrath_, Jan 01 2015

%C b(0) = 4; b(n+1) is the smallest number not in the sequence such that b(n+1) - Prod_{i=0..n} b(i) divides b(n+1) - Sum_{i=0..n} b(i). Then b(n) = a(n) for n > 2. - _Derek Orr_, Jan 15 2015

%C a(n) counts the permutations of length n+2 whose first element is 2 such that the permutation has exactly one descent. - _Ran Pan_, Apr 17 2015

%C a(0)-a(30) appear, with a(26)-a(30) in error, in tablet M 08613 (see CDLI link) from the Old Babylonian period (c. 1900-1600 BC). - _Charles R Greathouse IV_, Sep 03 2015

%C Subsequence of A028982 (the squares or twice squares sequence). - _Timothy L. Tiffin_, Jul 18 2016

%C A000120(a(n)) = 1. A000265(a(n)) = 1. A000593(a(n)) = 1. - _Juri-Stepan Gerasimov_, Aug 16 2016

%C Number of monotone maps f : [0..n] -> [0..n] which are order-increasing (i <= f(i)) and idempotent (f(f(i)) = f(i)). In other words, monads on the n-th ordinal (seen as a posetal category). Any monad f determines a subset of [0..n] that contains n, by considering its set of monad algebras = fixed points { i | f(i) = i }. Conversely, any subset S of [0..n] containing n determines a monad on [0..n], by the function i |-> min { j | i <= j, j in S }. - _Noam Zeilberger_, Dec 11 2016

%C Consider n points lying on a circle. Then for n>=2 a(n-2) gives the number of ways to connect two adjacent points with nonintersecting chords. - _Anton Zakharov_, Dec 31 2016

%C Satisfies Benford's law [Diaconis, 1977; Berger-Hill, 2017] - _N. J. A. Sloane_, Feb 07 2017

%C Also the number of independent vertex sets and vertex covers in the n-empty graph. - _Eric W. Weisstein_, Sep 21 2017

%C Also the number of maximum cliques in the n-halved cube graph for n > 4. - _Eric W. Weisstein_, Dec 04 2017

%D 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.

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

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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D V. E. Tarakanov, Combinatorial problems on binary matrices, Combin. Analysis, MSU, 5 (1980), 4-15. (Russian)

%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

%H N. J. A. Sloane, <a href="/A000079/b000079.txt">Table of n, 2^n for n = 0..1000</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://apps.nrbook.com/abramowitz_and_stegun/index.html">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Mohammad K. Azarian, <a href="http://www.math-cs.ucmo.edu/~mjms/2004.1/azar6.pdf">A Generalization of the Climbing Stairs Problem II</a>, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.

%H Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations on Integer Sequences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

%H A. Berger and T. P. Hill, <a href="http://www.ams.org/publications/journals/notices/201702/rnoti-p132.pdf">What is Benford's Law?</a>, Notices, Amer. Math. Soc., 64:2 (2017), 132-134.

%H Anicius Manlius Severinus Boethius, <a href="http://www.e-codices.unifr.ch/en/vad/0296/6r/medium">De arithmetica</a>, Book 1, section 9.

%H Henry Bottomley, <a href="/A000079/a000079.gif">Illustration of initial terms</a>

%H D. Butler, <a href="http://www.tsm-resources.com/alists/pow2.html">Powers of Two up to 2^222</a>

%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

%H P. J. Cameron, <a href="https://cameroncounts.wordpress.com/2017/05/15/notes-on-counting/">Notes on Counting</a>, Peter Cameron's Blog, 15/05/2017.

%H CDLI, <a href="http://cdli.ucla.edu/search/search_results.php?SearchMode=Text&amp;ObjectID=P390441">M 08613</a>.

%H Keneth Adrian P. Dagal and Jose Arnaldo B. Dris, <a href="http://arxiv.org/abs/1308.6767">A Criterion for Almost Perfect Numbers in Terms of the Abundancy Index</a>, arXiv:1308.6767v1 [math.NT], Aug 14 2013.

%H Persi Diaconis, <a href="https://doi.org/10.1214/aop/1176995891">The distribution of leading digits and uniform distribution mod 1</a>, Ann. Probability, 5, 1977, 72--81.

%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 18

%H Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Gewurz/gewurz5.html">Sequences realized as Parker vectors of oligomorphic permutation groups</a>, J. Integer Seqs., Vol. 6, 2003.

%H Hermann Gruber, Jonathan Lee and Jeffrey Shallit: <a href="http://arxiv.org/abs/1204.4982">Enumerating regular expressions and their languages</a>, arXiv:1204.4982v1 [cs.FL], 2012.

%H R. K. Guy, <a href="/A000346/a000346.pdf">Letter to N. J. A. Sloane</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=6">Encyclopedia of Combinatorial Structures 6</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=68">Encyclopedia of Combinatorial Structures 68</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=72">Encyclopedia of Combinatorial Structures 72</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=267">Encyclopedia of Combinatorial Structures 267</a>

%H Marcus Jaiclin, David DiRico, Christopher O'Sullivan and Stephen Tetreault, <a href="http://www.westfield.ma.edu/math/faculty/jaiclin/writings/research/pascals_triangle/">Pascal's Triangle Mod 2,3,5</a>

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>

%H J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.

%H P. A. MacMahon, <a href="http://www.jstor.org/stable/90632">Memoir on the Theory of the Compositions of Numbers</a>, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901.

%H Victor Meally, <a href="/A002868/a002868.pdf">Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.</a>

%H R. Ondrejka, <a href="http://www.jstor.org/stable/2004456">Exact values of 2^n, n=1(1)4000</a>, Math. Comp., 23 (1969), 456.

%H G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Pfeiffer/pfeiffer6.html">Counting Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

%H Robert Price, <a href="/A000079/a000079.txt">Comments on A000079 concerning Elementary Cellular Automata</a>, Feb 26 2016

%H S. Saito, T. Tanaka, N. Wakabayashi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Saito/saito22.html">Combinatorial Remarks on the Cyclic Sum Formula for Multiple Zeta Values </a>, J. Int. Seq. 14 (2011) # 11.2.4, Table 1

%H Michael Z. Spivey and Laura L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

%H J. Tanton, <a href="http://www.jstor.org/stable/25678324">A Dozen Questions about the Powers of Two</a>, Math Horizons, Vol. 8, pp. 5-10, September 2001.

%H G. Villemin's Almanac of Numbers, <a href="http://villemin.gerard.free.fr/Wwwgvmm/Nombre/Puiss2.htm">Puissances de 2</a>

%H Sage Weil, <a href="http://www.newdream.net/~sage/old/numbers/pow2.htm">1058 powers of two</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Abundance.html">Abundance</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BinomialSums.html">Binomial Sums</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BinomialTransform.html">Binomial Transform</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CollatzProblem.html">Hailstone Number (Collatz Problem)</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Composition.html">Composition</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EmptyGraph.html">Empty Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Erf.html">Erf</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FractionalPart.html">Fractional Part</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HalvedCubeGraph.html">Halved Cube Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Hypercube.html">Hypercube</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LeastDeficientNumber.html">Least Deficient Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximumClique.html">Maximum Clique</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerFractionalParts.html">PowerFractional Parts</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Subset.html">Subset</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VertexCover.html">Vertex Cover</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Almost_perfect_number">Almost perfect number</a>

%H S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (2).

%H <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>

%F a(n) = 2^n.

%F a(0) = 1; a(n) = 2*a(n-1).

%F G.f.: 1/(1-2*x).

%F E.g.f.: exp(2*x).

%F a(n)= Sum_{k=0..n} binomial(n, k).

%F 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, ...]. - _Philippe Deléham_, Feb 25 2004

%F n such that phi(n) = n/2, for n > 1, where phi is Euler's totient (A000010). - _Lekraj Beedassy_, Sep 07 2004

%F a(n) = StirlingS2(n+1,2) + 1. - _Ross La Haye_, Jan 09 2008

%F a(n+2)=6a(n+1)-8a(n), n=1,2,3,... with a(1)=1, a(2)=2. - _Yosu Yurramendi_, Aug 06 2008

%F a(n) = ka(n-1) + (4-2k)a(n-2) for any integer k and n > 1, with a(0) = 1, a(1) = 2. - _Jaume Oliver Lafont_, Dec 05 2008

%F 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. - _Thomas Wieder_, Feb 25 2009

%F a(0) = 1, a(1)=2; a(n)=a(n-1)^2/a(n-2), n>=2. - _Jaume Oliver Lafont_, Sep 22 2009

%F If p[i]=i-1 and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=det A. - _Milan Janjic_, May 02 2010

%F If p[i]=fibonacci(i-2) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=2, a(n-2)=det A. - _Milan Janjic_, May 08 2010

%F The sum of reciprocals, 1/1 + 1/2 + 1/4 + 1/8 + ... + 1/(2^n) + ... = 2. - _Mohammad K. Azarian_, Dec 29 2010

%F a(n) = 2*A001045(n) + A078008(n) = 3*A001045(n) + (-1)^n. - _Paul Barry_, Feb 20 2003

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

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

%F a(n) = A131577(n) + A011782(n) = A024495(n) + A131708(n) + A024493(n) = A000749(n) + A038503(n) + A038504(n) + A038505(n) = A139761(n) + A139748(n) + A139714(n) + A133476(n) + A139398(n). - _Paul Curtz_, Jul 25 2011

%F a(n) = row sums of A007318. - _Susanne Wienand_, Oct 21 2011

%F a(n) = Hypergeometric([-n],[],-1). - _Peter Luschny_, Nov 01 2011

%F G.f.: A(x)=B(x)/x, B(x) satisfies B(B(x))=x/(1-x)^2. - _Vladimir Kruchinin_, Nov 10 2011

%F a(n) = Sum_{k=0..=n} A201730(n,k)*(-1)^k. - _Philippe Deléham_, Dec 06 2011

%F 2^n = Sum_{k=1..floor(n/2 + 1/2)} C(n+1,2k-1). - _Dennis P. Walsh_, Dec 15 2011

%F Sum_{n>=1} mobius(n)/a(n) = 0.1020113348178103647430363939318... - _R. J. Mathar_, Aug 12 2012

%F E.g.f.: 1+2*x/(U(0)-x) where U(k)= 6*k+1 + x^2/(6*k+3 + x^2/(6*k+5 + x^2/U(k+1) )); (continued fraction, 3-step). - _Sergei N. Gladkovskii_, Dec 04 2012

%F a(n) = det(|s(i+2,j)|, 1 <= i,j <= n), where s(n,k) are Stirling numbers of the first kind. - _Mircea Merca_, Apr 04 2013

%F a(n) = det(|ps(i+1,j)|, 1 <= i,j <= n), where ps(n,k) are Legendre-Stirling numbers of the first kind (A129467). - _Mircea Merca_, Apr 06 2013

%F G.f.: W(0), where W(k) = 1 + 2*x*(k+1)/(1 - 2*x*(k+1)/( 2*x*(k+2) + 1/W(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 28 2013

%F a(n-1) = sum_{t_1 + 2*t_2 + ... + n*t_n = n} multinomial(t_1 + t_2 + ... + t_n, t_1, t_2, ..., t_n)). - _Mircea Merca_, Dec 06 2013

%F Construct the power matrix T(n,j)=[A^*j]*[S^*(j-1)] where A(n)=(1,1,1,...) and S(n)=(0,1,0,0...) (where * is convolution operation). Then a(n-1) = sum_{j=1..n} T(n,j). - _David Neil McGrath_, Jan 01 2015

%F a(n) = A000005(A002110(n)). - _Ivan N. Ianakiev_, May 23 2016

%F From _Ilya Gutkovskiy_, Jul 18 2016: (Start)

%F Exponential convolution of A000012 with themselves.

%F a(n) = Sum_{k=0..n} A011782(k).

%F Sum_{n>=0} a(n)/n! = exp(2) = A072334.

%F Sum_{n>=0} (-1)^n*a(n)/n! = exp(-2) = A092553. (End)

%F G.f.: (r(x) * r(x^2) * r(x^4) * r(x^8) * ...) where r(x) = A090129(x) = (1 + 2x + 2x^2 + 4x^3 + 8x^4 + ...). - _Gary W. Adamson_, Sep 13 2016

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

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

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

%p 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 ); # _Zerinvary Lajos_, Apr 05 2009

%p isA000079 := proc(n)

%p local fs;

%p fs := numtheory[factorset](n) ;

%p if n = 1 then

%p true ;

%p elif nops(fs) <> 1 then

%p false;

%p elif op(1,fs) = 2 then

%p true;

%p else

%p false ;

%p end if;

%p end proc: # _R. J. Mathar_, Jan 09 2017

%t Table[2^n, {n, 0, 50}]

%t 2^Range[0, 50] (* _Wesley Ivan Hurt_, Jun 14 2014 *)

%t LinearRecurrence[{2}, {2}, {0, 20}] (* _Eric W. Weisstein_, Sep 21 2017 *)

%t CoefficientList[Series[1/(1 - 2 x), {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 21 2017 *)

%o (PARI) A000079(n)=2^n \\ Edited by _M. F. Hasler_, Aug 27 2014

%o (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

%o (PARI) x=1; for (n=0, 1000, write("b000079.txt", n, " ", x); x+=x); \\ _Harry J. Smith_, Apr 26 2009

%o (Haskell)

%o a000079 = (2 ^)

%o a000079_list = iterate (* 2) 1

%o -- _Reinhard Zumkeller_, Jan 22 2014, Mar 05 2012, Dec 29 2011

%o (Maxima) A000079(n):=2^n$ makelist(A000079(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */

%o (MAGMA) [2^n: n in [0..40]] (* or *) [n le 2 select n else 5*Self(n-1)-6*Self(n-2): n in [1..40]]; // _Vincenzo Librandi_, Feb 17 2014

%o (Scheme) (define (A000079 n) (expt 2 n)) ;; _Antti Karttunen_, Mar 21 2017

%Y Subsequence of A028982.

%Y Cf. A000225, A038754, A133464, A140730, A037124, A001787, A001788, A001789, A003472, A054849, A002409, A054851, A140325, A140354, A000041, A152537, A001405, A007318, A000120, A000265, A000593, A001227.

%Y This is the Hankel transform (see A001906 for the definition) of A000984, A002426, A026375, A026387, A026569, A026585, A026671 and A032351. - _John W. Layman_, Jul 31 2000

%Y Euler transform of A001037, inverse binomial transform of A000244, binomial transform of A000012.

%Y Complement of A057716.

%Y Boustrophedon transforms: A000734, A000752.

%Y Range of values of A006519, A007875, A011782, A030001, A034444, A037445, A053644, and A054243.

%Y Cf. A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691 (sum of 2, ..., 9 distinct powers of 2).

%Y Cf. A090129.

%K nonn,core,easy,nice,changed

%O 0,2

%A _N. J. A. Sloane_, Apr 30 1991

%E Clarified a comment _T. D. Noe_, Aug 30 2009

%E Edited by _Daniel Forgues_, May 12 2010

%E Incorrect comment deleted by _Matthew Vandermast_, May 17 2014

%E Comment corrected to match offset by _Geoffrey Critzer_, Nov 28 2014

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 18 14:30 EST 2017. Contains 296177 sequences.