cp's OEIS Frontend

A000079 Powers of 2: a(n) = 2^n.

%I A000079 M1129 N0432 #874 Aug 15 2025 21:14:45
%S A000079 1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536,
%T A000079 131072,262144,524288,1048576,2097152,4194304,8388608,16777216,
%U A000079 33554432,67108864,134217728,268435456,536870912,1073741824,2147483648,4294967296,8589934592
%N A000079 Powers of 2: a(n) = 2^n.
%C A000079 2^0 = 1 is the only odd power of 2.
%C A000079 Number of subsets of an n-set.
%C A000079 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 A000079 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 A000079 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 A000079 a(n+1) is the smallest number that is not the sum of any number of (distinct) earlier terms.
%C A000079 Same as Pisot sequences E(1, 2), L(1, 2), P(1, 2), T(1, 2). See A008776 for definitions of Pisot sequences.
%C A000079 With initial 1 omitted, same as Pisot sequences E(2, 4), L(2, 4), P(2, 4), T(2, 4). - _David W. Wilson_
%C A000079 Not the sum of two or more consecutive numbers. - _Lekraj Beedassy_, May 14 2004
%C A000079 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 A000079 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 A000079 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 A000079 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 A000079 The only hailstone sequence which doesn't rebound (except "on the ground"). - _Alexandre Wajnberg_, Jan 29 2005
%C A000079 With p(n) as the number of integer partitions of n, p(i) is the number of parts of the i-th partition of n, d(i) is the number of different parts of the i-th partition of n, m(i,j) is the multiplicity of the j-th part of the i-th partition of n, one has: a(n) = Sum_{i = 1..p(n)} (p(i)! / (Product_{j=1..d(i)} m(i,j)!)). - _Thomas Wieder_, May 18 2005
%C A000079 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 A000079 The first differences are the sequence itself. - _Alexandre Wajnberg_ and _Eric Angelini_, Sep 07 2005
%C A000079 a(n) is the largest number with shortest addition chain involving n additions. - _David W. Wilson_, Apr 23 2006
%C A000079 Beginning with a(1) = 0, numbers not equal to the sum of previous distinct natural numbers. - _Giovanni Teofilatto_, Aug 06 2006
%C A000079 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 A000079 Let P(A) be the power set of an n-element set A. Then a(n) is the number of pairs of elements {x,y} of P(A) for which x = y. - _Ross La Haye_, Jan 09 2008
%C A000079 a(n) is the 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 A000079 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 A000079 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 A000079 Successive k such that phi(k)/k = 1/2, where phi is Euler's totient function. - _Artur Jasinski_, Nov 07 2008
%C A000079 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 A000079 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 A000079 a(n) = a(n-1)-th even natural number (A005843) for n > 1. - _Jaroslav Krizek_, Apr 25 2009
%C A000079 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 A000079 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 A000079 Catalan transform of A099087. - _R. J. Mathar_, Jun 29 2009
%C A000079 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 A000079 Or, phi(n) is equal to the number of perfect partitions of n. - _Juri-Stepan Gerasimov_, Oct 10 2009
%C A000079 These are the 2-smooth numbers, positive integers with no prime factors greater than 2. - _Michael B. Porter_, Oct 04 2009
%C A000079 A064614(a(n)) = A000244(n) and A064614(m) < A000244(n) for m < a(n). - _Reinhard Zumkeller_, Feb 08 2010
%C A000079 a(n) is 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 A000079 a(n) is the smallest proper multiple of a(n-1). - _Dominick Cancilla_, Aug 09 2010
%C A000079 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 A000079 Records in the number of prime factors. - _Juri-Stepan Gerasimov_, Mar 12 2011
%C A000079 Row sums of A152538. - _Gary W. Adamson_, Dec 10 2008
%C A000079 A078719(a(n)) = 1; A006667(a(n)) = 0. - _Reinhard Zumkeller_, Oct 08 2011
%C A000079 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 A000079 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 A000079 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_{k=1..floor((n+1)/2)} C(n+1, 2k-1). - _Dennis P. Walsh_, Dec 15 2011
%C A000079 a(n) is the 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 A000079 A204455(k) = 1 if and only if k is in this sequence. - _Wolfdieter Lang_, Feb 04 2012
%C A000079 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 A000079 First differences of A000225. - _Omar E. Pol_, Feb 19 2013
%C A000079 This is the lexicographically earliest sequence which contains no arithmetic progression of length 3. - Daniel E. Frohardt, Apr 03 2013
%C A000079 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 A000079 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 A000079 More is known now about non-power-of-2 "Almost Perfect Numbers" as described in Dagal. - _Jonathan Vos Post_, Sep 01 2013
%C A000079 Number of symmetric Ferrers diagrams that fit into an n X n box. - _Graham H. Hawkes_, Oct 18 2013
%C A000079 Numbers n such that sigma(2n) = 2n + sigma(n). - _Jahangeer Kholdi_, Nov 23 2013
%C A000079 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 A000079 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 A000079 A072219(a(n)) = 1. - _Reinhard Zumkeller_, Feb 20 2014
%C A000079 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 A000079 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 A000079 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 A000079 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 A000079 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 A000079 Numbers n such that sigma(n) = sigma(2n) - phi(4n). - _Farideh Firoozbakht_, Aug 14 2014
%C A000079 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 A000079 a(n) counts n-walks (closed) on the graph G(1-vertex; 1-loop, 1-loop). - _David Neil McGrath_, Dec 11 2014
%C A000079 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 A000079 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 A000079 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 A000079 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 A000079 Subsequence of A028982 (the squares or twice squares sequence). - _Timothy L. Tiffin_, Jul 18 2016
%C A000079 A000120(a(n)) = 1. A000265(a(n)) = 1. A000593(a(n)) = 1. - _Juri-Stepan Gerasimov_, Aug 16 2016
%C A000079 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 A000079 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 A000079 Satisfies Benford's law [Diaconis, 1977; Berger-Hill, 2017] - _N. J. A. Sloane_, Feb 07 2017
%C A000079 Also the number of independent vertex sets and vertex covers in the n-empty graph. - _Eric W. Weisstein_, Sep 21 2017
%C A000079 Also the number of maximum cliques in the n-halved cube graph for n > 4. - _Eric W. Weisstein_, Dec 04 2017
%C A000079 Number of pairs of compositions of n corresponding to a seaweed algebra of index n-1. - _Nick Mayers_, Jun 25 2018
%C A000079 The multiplicative group of integers modulo a(n) is cyclic if and only if n = 0, 1, 2. For n >= 3, it is a product of two cyclic groups. - _Jianing Song_, Jun 27 2018
%C A000079 k^n is the determinant of n X n matrix M_(i, j) = binomial(k + i + j - 2, j) - binomial(i+j-2, j), in this case k=2. - _Tony Foster III_, May 12 2019
%C A000079 Solutions to the equation Phi(2n + 2*Phi(2n)) = 2n. - _M. Farrokhi D. G._, Jan 03 2020
%C A000079 a(n-1) is the number of subsets of {1,2,...,n} which have an element that is the size of the set. For example, for n = 4, a(3) = 8 and the subsets are {1}, {1,2}, {2,3}, {2,4}, {1,2,3}, {1,3,4}, {2,3,4}, {1,2,3,4}. - _Enrique Navarrete_, Nov 21 2020
%C A000079 a(n) is the number of self-inverse (n+1)-order permutations with 231-avoiding. E.g., a(3) = 8: [1234, 1243, 1324, 1432, 2134, 2143, 3214, 4321]. - _Yuchun Ji_, Feb 26 2021
%C A000079 For any fixed k > 0, a(n) is the number of ways to tile a strip of length n+1 with tiles of length 1, 2, ... k, where the tile of length k can be black or white, with the restriction that the first tile cannot be black. - _Greg Dresden_ and Bora Bursalı, Aug 31 2023
%D A000079 Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 1016.
%D A000079 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 A000079 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 73, 84.
%D A000079 Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §4.5 Logarithms and §8.1 Terminology, pp. 150, 264.
%D A000079 Paul J. Nahin, An Imaginary Tale: The Story of sqrt(-1), Princeton University Press, Princeton, NJ. 1998, pp. 69-70.
%D A000079 Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 273.
%D A000079 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 124.
%D A000079 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000079 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000079 V. E. Tarakanov, Combinatorial problems on binary matrices, Combin. Analysis, MSU, 5 (1980), 4-15. (Russian)
%D A000079 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 141.
%D A000079 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H A000079 N. J. A. Sloane, <a href="/A000079/b000079.txt">Table of n, 2^n for n = 0..1000</a>
%H A000079 Milton Abramowitz and Irene A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A000079 Juan S. Auli and Sergi Elizalde, <a href="https://arxiv.org/abs/2003.11533">Wilf equivalences between vincular patterns in inversion sequences</a>, arXiv:2003.11533 [math.CO], 2020.
%H A000079 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 A000079 Jonathan Beagley and Lara Pudwell, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Pudwell/pudwell13.html">Colorful Tilings and Permutations</a>, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.4.
%H A000079 Arno Berger and Theodore 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 A000079 Tobias Boege and Thomas Kahle, <a href="https://arxiv.org/abs/1902.11260">Construction Methods for Gaussoids</a>, arXiv:1902.11260 [math.CO], 2019.
%H A000079 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 A000079 Henry Bottomley, <a href="/A000079/a000079.gif">Illustration of initial terms</a>
%H A000079 Douglas Butler, <a href="https://web.archive.org/web/20191202164323/http://www.tsm-resources.com:80/alists/pow2.html">Powers of Two up to 2^222</a>
%H A000079 Peter 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 A000079 Peter 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 A000079 CDLI, <a href="http://cdli.ucla.edu/search/search_results.php?SearchMode=Text&amp;ObjectID=P390441">M 08613</a>.
%H A000079 Giulio Cerbai, Anders Claesson, and Luca Ferrari, <a href="https://arxiv.org/abs/1907.08142">Stack sorting with restricted stacks</a>, arXiv:1907.08142 [cs.DS], 2019.
%H A000079 V. Coll, M. Hyatt, C. Magnant, and H. Wang, <a href="http://dx.doi.org/10.4172/1736-4337.1000227">Meander graphs and Frobenius seaweed Lie algebras II</a>, Journal of Generalized Lie Theory and Applications 9 (1) (2015) 227.
%H A000079 M. Coons and H. Winning, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Coons/coons2.html">Powers of Two Modulo Powers of Three</a>, J. Int. Seq. 18 (2015) # 15.6.1.
%H A000079 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 A000079 V. Dergachev and A. Kirillov, <a href="https://www.emis.de/journals/JLT/vol.10_no.2/6.html">Index of Lie algebras of seaweed type</a>, J. Lie Theory 10 (2) (2000) 331-343.
%H A000079 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 A000079 Kenneth Edwards and Michael A. Allen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Allen/edwards2.html">New combinatorial interpretations of the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers using two types of tile</a>, J. Int. Seq. 24 (2021) Article 21.3.8.
%H A000079 David Eppstein, <a href="https://arxiv.org/abs/1804.07396">Making Change in 2048</a>, arXiv:1804.07396 [cs.DM], 2018.
%H A000079 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 18
%H A000079 Joël Gay and Vincent Pilaud, <a href="https://arxiv.org/abs/1804.06572">The weak order on Weyl posets</a>, arXiv:1804.06572 [math.CO], 2018.
%H A000079 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 A000079 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 A000079 R. K. Guy, <a href="/A000346/a000346.pdf">Letter to N. J. A. Sloane</a>
%H A000079 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=6">Encyclopedia of Combinatorial Structures 6</a>
%H A000079 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=68">Encyclopedia of Combinatorial Structures 68</a>
%H A000079 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=72">Encyclopedia of Combinatorial Structures 72</a>
%H A000079 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=267">Encyclopedia of Combinatorial Structures 267</a>
%H A000079 Marcus Jaiclin, et al. <a href="https://web.archive.org/web/20170823000349/http://pyrrho.wsc.ma.edu/math/faculty/jaiclin/writings/research/pascals_triangle/">Pascal's Triangle, Mod 2,3,5</a>
%H A000079 Milan Janjic, <a href="https://old.pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>
%H A000079 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 A000079 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 A000079 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 A000079 Augustine O. Munagi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Munagi/munagi10.html">Integer Compositions and Higher-Order Conjugation</a>, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
%H A000079 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 A000079 Permutation Pattern Avoidance Library (PermPAL), <a href="https://permpal.com/perms/basis/012_021/">Av(012_021)</a>
%H A000079 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 A000079 Robert Price, <a href="/A000079/a000079.txt">Comments on A000079 concerning Elementary Cellular Automata</a>, Feb 26 2016.
%H A000079 Shingo Saito, Tatsushi Tanaka, and Noriko 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 A000079 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 A000079 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 A000079 G. Villemin's Almanac of Numbers, <a href="http://villemin.gerard.free.fr/Wwwgvmm/Nombre/Puiss2.htm">Puissances de 2</a>
%H A000079 Sage Weil, <a href="https://web.archive.org/web/20090206092940/http://newdream.net:80/~sage/old/numbers/pow2.htm">1058 powers of two</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Abundance.html">Abundance</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BinomialSums.html">Binomial Sums</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BinomialTransform.html">Binomial Transform</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CollatzProblem.html">Hailstone Number (Collatz Problem)</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Composition.html">Composition</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EmptyGraph.html">Empty Graph</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Erf.html">Erf</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FractionalPart.html">Fractional Part</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HalvedCubeGraph.html">Halved Cube Graph</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Hypercube.html">Hypercube</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LeastDeficientNumber.html">Least Deficient Number</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximumClique.html">Maximum Clique</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerFractionalParts.html">PowerFractional Parts</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Subset.html">Subset</a>
%H A000079 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/VertexCover.html">Vertex Cover</a>
%H A000079 Wikipedia, <a href="http://en.wikipedia.org/wiki/Almost_perfect_number">Almost perfect number</a>
%H A000079 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A000079 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A000079 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H A000079 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A000079 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H A000079 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H A000079 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%H A000079 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (2).
%H A000079 <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>
%F A000079 a(n) = 2^n.
%F A000079 a(0) = 1; a(n) = 2*a(n-1).
%F A000079 G.f.: 1/(1 - 2*x).
%F A000079 E.g.f.: exp(2*x).
%F A000079 a(n)= Sum_{k = 0..n} binomial(n, k).
%F A000079 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 A000079 n such that phi(n) = n/2, for n > 1, where phi is Euler's totient (A000010). - _Lekraj Beedassy_, Sep 07 2004
%F A000079 a(n + 1) = a(n) XOR 3*a(n) where XOR is the binary exclusive OR operator. - _Philippe Deléham_, Jun 19 2005
%F A000079 a(n) = StirlingS2(n + 1, 2) + 1. - _Ross La Haye_, Jan 09 2008
%F A000079 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 A000079 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 A000079 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 A000079 a(0) = 1, a(1) = 2; a(n) = a(n-1)^2/a(n-2), n >= 2. - _Jaume Oliver Lafont_, Sep 22 2009
%F A000079 a(n) = A173786(n, n)/2 = A173787(n + 1, n). - _Reinhard Zumkeller_, Feb 28 2010
%F A000079 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 A000079 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 A000079 The sum of reciprocals, 1/1 + 1/2 + 1/4 + 1/8 + ... + 1/(2^n) + ... = 2. - _Mohammad K. Azarian_, Dec 29 2010
%F A000079 a(n) = 2*A001045(n) + A078008(n) = 3*A001045(n) + (-1)^n. - _Paul Barry_, Feb 20 2003
%F A000079 a(n) = A118654(n, 2).
%F A000079 a(n) = A140740(n+1, 1).
%F A000079 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 A000079 a(n) = row sums of A007318. - _Susanne Wienand_, Oct 21 2011
%F A000079 a(n) = Hypergeometric([-n], [], -1). - _Peter Luschny_, Nov 01 2011
%F A000079 G.f.: A(x) = B(x)/x, B(x) satisfies B(B(x)) = x/(1 - x)^2. - _Vladimir Kruchinin_, Nov 10 2011
%F A000079 a(n) = Sum_{k = 0..n} A201730(n, k)*(-1)^k. - _Philippe Deléham_, Dec 06 2011
%F A000079 2^n = Sum_{k = 1..floor((n+1)/2)} C(n+1, 2k-1). - _Dennis P. Walsh_, Dec 15 2011
%F A000079 A209229(a(n)) = 1. - _Reinhard Zumkeller_, Mar 07 2012
%F A000079 A001227(a(n)) = 1. - _Reinhard Zumkeller_, May 01 2012
%F A000079 Sum_{n >= 1} mobius(n)/a(n) = 0.1020113348178103647430363939318... - _R. J. Mathar_, Aug 12 2012
%F A000079 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 A000079 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 A000079 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 A000079 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 A000079 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 A000079 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 A000079 a(n) = A000005(A002110(n)). - _Ivan N. Ianakiev_, May 23 2016
%F A000079 From _Ilya Gutkovskiy_, Jul 18 2016: (Start)
%F A000079 Exponential convolution of A000012 with themselves.
%F A000079 a(n) = Sum_{k=0..n} A011782(k).
%F A000079 Sum_{n>=0} a(n)/n! = exp(2) = A072334.
%F A000079 Sum_{n>=0} (-1)^n*a(n)/n! = exp(-2) = A092553. (End)
%F A000079 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
%F A000079 a(n) = A000045(n + 1) + A000045(n) + Sum_{k = 0..n - 2} A000045(k + 1)*2^(n - 2 - k). - _Melvin Peralta_, Dec 22 2017
%F A000079 a(n) = 7*A077020(n)^2 + A077021(n)^2, n>=3. - _Ralf Steiner_, Aug 08 2021
%F A000079 a(n)= n + 1 + Sum_{k=3..n+1} (2*k-5)*J(n+2-k), where Jacobsthal number J(n) = A001045(n). - _Michael A. Allen_, Jan 12 2022
%F A000079 Integral_{x=0..Pi} cos(x)^n*cos(n*x) dx = Pi/a(n) (see Nahin, pp. 69-70). - _Stefano Spezia_, May 17 2023
%e A000079 There are 2^3 = 8 subsets of a 3-element set {1,2,3}, namely { -, 1, 2, 3, 12, 13, 23, 123 }.
%p A000079 A000079 := n->2^n; [ seq(2^n,n=0..50) ];
%p A000079 isA000079 := proc(n)
%p A000079     local fs;
%p A000079     fs := numtheory[factorset](n) ;
%p A000079     if n = 1 then
%p A000079         true ;
%p A000079     elif nops(fs) <> 1 then
%p A000079         false;
%p A000079     elif op(1,fs) = 2 then
%p A000079         true;
%p A000079     else
%p A000079         false ;
%p A000079     end if;
%p A000079 end proc: # _R. J. Mathar_, Jan 09 2017
%t A000079 Table[2^n, {n, 0, 50}]
%t A000079 2^Range[0, 50] (* _Wesley Ivan Hurt_, Jun 14 2014 *)
%t A000079 LinearRecurrence[{2}, {2}, {0, 20}] (* _Eric W. Weisstein_, Sep 21 2017 *)
%t A000079 CoefficientList[Series[1/(1 - 2 x), {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 21 2017 *)
%t A000079 NestList[2# &, 1, 40] (* _Harvey P. Dale_, Oct 07 2019 *)
%o A000079 (PARI) A000079(n)=2^n \\ Edited by _M. F. Hasler_, Aug 27 2014
%o A000079 (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 A000079 (Haskell)
%o A000079 a000079 = (2 ^)
%o A000079 a000079_list = iterate (* 2) 1
%o A000079 -- _Reinhard Zumkeller_, Jan 22 2014, Mar 05 2012, Dec 29 2011
%o A000079 (Maxima) A000079(n):=2^n$ makelist(A000079(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */
%o A000079 (Magma) [2^n: n in [0..40]]; // _Vincenzo Librandi_, Feb 17 2014
%o A000079 (Magma) [n le 2 select n else 5*Self(n-1)-6*Self(n-2): n in [1..40]]; // _Vincenzo Librandi_, Feb 17 2014
%o A000079 (Scheme) (define (A000079 n) (expt 2 n)) ;; _Antti Karttunen_, Mar 21 2017
%o A000079 (Scala) (List.fill(20)(2: BigInt)).scanLeft(1: BigInt)(_ * _) // _Alonso del Arte_, Jan 16 2020
%o A000079 (Python)
%o A000079 def a(n): return 1<<n
%o A000079 print([a(n) for n in range(34)]) # _Michael S. Branicky_, Jul 28 2022
%o A000079 (Python)
%o A000079 def is_powerof2(n) -> bool: return n and (n & (n - 1)) == 0  # _Peter Luschny_, Apr 10 2025
%Y A000079 Cf. A000225, A038754, A133464, A140730, A037124, A001787, A001788, A001789, A003472, A054849, A002409, A054851, A140325, A140354, A000041, A152537, A001405, A007318, A000120, A000265, A000593, A001227, A077020, A077021.
%Y A000079 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 A000079 Euler transform of A001037, A209406 (multisets), inverse binomial transform of A000244, binomial transform of A000012.
%Y A000079 Complement of A057716.
%Y A000079 Boustrophedon transforms: A000734, A000752.
%Y A000079 Range of values of A006519, A007875, A011782, A030001, A034444, A037445, A053644, and A054243.
%Y A000079 Cf. A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691 (sum of 2, ..., 9 distinct powers of 2).
%Y A000079 Cf. A090129.
%Y A000079 The following are parallel families: A000079 (2^n), A004094 (2^n reversed), A028909 (2^n sorted up), A028910 (2^n sorted down), A036447 (double and reverse), A057615 (double and sort up), A263451 (double and sort down); A000244 (3^n), A004167 (3^n reversed), A321540 (3^n sorted up), A321539 (3^n sorted down), A163632 (triple and reverse), A321542 (triple and sort up), A321541 (triple and sort down).
%K A000079 nonn,core,easy,nice
%O A000079 0,2
%A A000079 _N. J. A. Sloane_
%E A000079 Clarified a comment _T. D. Noe_, Aug 30 2009
%E A000079 Edited by _Daniel Forgues_, May 12 2010
%E A000079 Incorrect comment deleted by _Matthew Vandermast_, May 17 2014
%E A000079 Comment corrected to match offset by _Geoffrey Critzer_, Nov 28 2014