Search: id:A000079 Results 1-1 of 1 results found. %I A000079 M1129 N0432 %S A000079 1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536,131072, %T A000079 262144,524288,1048576,2097152,4194304,8388608,16777216,33554432,67108864, %U A000079 134217728,268435456,536870912,1073741824,2147483648,4294967296,8589934592 %N A000079 Powers of 2: a(n) = 2^n. %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 analogue of the preferential labelings sequence A000670. %C A000079 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. %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 (njas(AT)research.att.com), Oct 26, 2003. %C A000079 a(n+1) = 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 (blekraj(AT)yahoo.com), May 14 2004 %C A000079 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. %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 only hailstone sequence which doesn't rebound (except "on the ground"). - Alexandre Wajnberg (alexandre.wajnberg(AT)ulb.ac.be), Jan 29 2005 %C A000079 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 %C A000079 a(n+1) = a(n) XOR 3a(n) where XOR is binary exclusive OR operator. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 19 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 An autocopy sequence: its first differences are the sequence itself. - Alexandre Wajnberg & Eric Angelini (alexandre.wajnberg(AT)ulb.ac.be), Sep 07 2005 %C A000079 a(n) = largest number with shortest addition chain involving n additions. - David W. Wilson (davidwwilson(AT)comcast.net), Apr 23 2006 %C A000079 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 %C A000079 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 %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 R. Janjic (agnus(AT)blic.net), Mar 27 2007 %C A000079 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 %C A000079 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 %C A000079 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 %C A000079 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] %C A000079 Contribution from Bill R McEachen (bmceachen(AT)centralsan.org), Oct 29 2008: (Start) %C A000079 a(n) appears to match the number of divisors of the modified primorials (excluding 2,3and 5) %C A000079 Very limited range examined, PARI example shown (End) %C A000079 Successive k such that EulerPhi[k]/k = 1/2. [From Artur Jasinski (grafix(AT)csl.pl), 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. [From Paul Curtz (bpcrtz(AT)free.fr), Jan 05 2009] %C A000079 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] %C A000079 a(n) = a(n-1)-th even natural numbers (A005843) for n > 1. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), 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. [From Kailasam Viswanathan Iyer (kvi(AT)nitt.edu), May 04 2009] %C A000079 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] %C A000079 Catalan transform of A099087. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 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)). [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Aug 02 2009] %C A000079 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] %C A000079 Or, phi(n) is equal to the number of perfect partitions of n. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 10 2009] %C A000079 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] %D A000079 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 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 Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5. %D A000079 R. Ondrejka, Exact values of 2^n, n=1(1)4000, Math. Comp., 23 (1969), 456. %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 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. %H A000079 N. J. A. Sloane, Table of n, 2^n for n = 0..1000 %H A000079 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]. %H A000079 Henry Bottomley, Illustration of initial terms %H A000079 D. Butler, Powers of Two up to 2^222 %H A000079 P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. %H A000079 P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 18 %H A000079 Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors ..., J. Integer Seqs., Vol. 6, 2003. %H A000079 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 6 %H A000079 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 68 %H A000079 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 72 %H A000079 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 267 %H A000079 Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets %H A000079 J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5. %H A000079 G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2. %H A000079 G. Villemin's Almanac of Numbers, Puissances de 2 %H A000079 Sage Weil, 1058 powers of two %H A000079 Eric Weisstein's World of Mathematics, Fractional Part %H A000079 Eric Weisstein's World of Mathematics, PowerFractional Parts %H A000079 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (1). %H A000079 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (2). %H A000079 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (3). %H A000079 Eric Weisstein's World of Mathematics, Hypercube %H A000079 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics(4) %H A000079 Eric Weisstein's World of Mathematics, Hailstone Number %H A000079 Eric Weisstein's World of Mathematics, Erf %H A000079 Eric Weisstein's World of Mathematics, Abundance %H A000079 Wikipedia, Almost perfect number %H A000079 Index entries for "core" sequences %H A000079 Index entries for related partition-counting sequences %H A000079 Index entries for sequences related to linear recurrences with constant coefficients %F A000079 a(n) = 2^n; a(n) = 2*a(n-1). G.f.: 1/(1-2x), e.g.f.: exp(2x). %F A000079 2^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, ...]. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 25 2004 %F A000079 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 %F A000079 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 %F A000079 a(n) = StirlingS2(n+1,2) + 1 - Ross La Haye (rlahaye(AT)new.rr.com), Jan 09 2008 Ross La Haye %F A000079 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 %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. [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Dec 05 2008] %F A000079 Equals the partition numbers A000041 convolved with A152537. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 06 2008] %F A000079 Formula from Thomas Wieder (wieder.thomas(AT)t-online.de), Feb 25 2009: %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} %F A000079 delta(l_1,l_2,...,l_i,...,l_n) %F A000079 where delta(l_1,l_2,...,l_i,...,l_n) = 0 if any l_i <= l_(i+1) and l_(i+1) <> 0 %F A000079 and delta(l_1,l_2,...,l_i,...,l_n) = 1 otherwise. %F A000079 G.f.: exp(x)*cosh(x) . [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 05 2009] %F A000079 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] %F A000079 A000010(a(n))=A002033(a(n)). [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 10 2009] %e A000079 There are 2^3 = 8 subsets of a 3-element set {1,2,3}, namely { -, 1, 2, 3, 12, 13, 23, 123 }. %e A000079 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] %p A000079 A000079 := n->2^n; [ seq(2^n,n=0..50) ]; %p A000079 with(combstruct); SeqSetU := [S, {S=Sequence(U), U=Set(Z,card >= 1)}, unlabeled]; seq(count(SeqSetU, size=j),j=1..12); %p A000079 with(combinat):seq(stirling2(n,2)+1, n=1..34); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 04 2007 %p A000079 seq(binomial(n+0,0)*2^n,n=0..33); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 08 2008 %p A000079 with(finance):seq(futurevalue(2,1,n), n=-1..31);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 24 2009] %p A000079 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] %t A000079 Array[ 2^#&, 50, 0 ] %t A000079 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] %o A000079 (PARI) a(n)=if(n<0,0,2^n) %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,um=0); if (um==0,break)); if (um==1,print(x)); umc+=um); umc } %o A000079 sage: [lucas_number2(n,4,4) for n in xrange(-1,27)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 08 2008 %o A000079 (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] %o A000079 (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] %Y A000079 a(n) = 2*A001045(n)+A078008(n) = 3*A001045(n)+(-1)^n. - Paul Barry (pbarry(AT)wit.ie), Feb 20 2003 %Y A000079 Cf. A000225. %Y A000079 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 %Y A000079 Euler transform of A001037. %Y A000079 Complement of A057716. %Y A000079 a(n) = A118654(n, 2). %Y A000079 a(n) = A140740(n+1, 1). %Y A000079 Cf. A038754, A133464, A140730, A037124. %Y A000079 Cf. A001787, A001788, A001789, A003472, A054849, A002409, A054851, A140325, A140354. %Y A000079 Cf. A000041, A152537 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 06 2008] %Y A000079 Equals row sums of the partition convolution triangle, A152538 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 10 2008] %Y A000079 Cf. A000010, A002033.[From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 10 2009] %Y A000079 Sequence in context: A166444 A084633 A122803 this_sequence A120617 A050732 A138815 %Y A000079 Adjacent sequences: A000076 A000077 A000078 this_sequence A000080 A000081 A000082 %K A000079 core,easy,nice,nonn %O A000079 0,2 %A A000079 N. J. A. Sloane (njas(AT)research.att.com). %E A000079 Clarified a comment T. D. Noe (noe(AT)sspectra.com), Aug 30 2009 Search completed in 0.006 seconds