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%I M2655 N1059 #788 Apr 13 2024 22:58:32
%S 0,1,3,7,15,31,63,127,255,511,1023,2047,4095,8191,16383,32767,65535,
%T 131071,262143,524287,1048575,2097151,4194303,8388607,16777215,
%U 33554431,67108863,134217727,268435455,536870911,1073741823,2147483647,4294967295,8589934591
%N a(n) = 2^n - 1. (Sometimes called Mersenne numbers, although that name is usually reserved for A001348.)
%C This is the Gaussian binomial coefficient [n,1] for q=2.
%C Number of rank-1 matroids over S_n.
%C Numbers k such that the k-th central binomial coefficient is odd: A001405(k) mod 2 = 1. - _Labos Elemer_, Mar 12 2003
%C This gives the (zero-based) positions of odd terms in the following convolution sequences: A000108, A007460, A007461, A007463, A007464, A061922.
%C Also solutions (with minimum number of moves) for the problem of Benares Temple, i.e., three diamond needles with n discs ordered by decreasing size on the first needle to place in the same order on the third one, without ever moving more than one disc at a time and without ever placing one disc at the top of a smaller one. - Xavier Acloque, Oct 18 2003
%C a(0) = 0, a(1) = 1; a(n) = smallest number such that a(n)-a(m) == 0 (mod (n-m+1)), for all m. - _Amarnath Murthy_, Oct 23 2003
%C Binomial transform of [1, 1/2, 1/3, ...] = [1/1, 3/2, 7/3, ...]; (2^n - 1)/n, n=1,2,3, ... - _Gary W. Adamson_, Apr 28 2005
%C Numbers whose binary representation is 111...1. E.g., the 7th term is (2^7) - 1 = 127 = 1111111 (in base 2). - _Alexandre Wajnberg_, Jun 08 2005
%C Number of nonempty subsets of a set with n elements. - _Michael Somos_, Sep 03 2006
%C For n >= 2, a(n) is the least Fibonacci n-step number that is not a power of 2. - _Rick L. Shepherd_, Nov 19 2007
%C Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements {x,y} of P(A) for which x and y are disjoint and for which either x is a subset of y or y is a subset of x. - _Ross La Haye_, Jan 10 2008
%C A simpler way to state this is that it is the number of pairs (x,y) where at least one of x and y is the empty set. - _Franklin T. Adams-Watters_, Oct 28 2011
%C 2^n-1 is the sum of the elements in a Pascal triangle of depth n. - Brian Lewis (bsl04(AT)uark.edu), Feb 26 2008
%C Sequence generalized: a(n) = (A^n -1)/(A-1), n >= 1, A integer >= 2. This sequence has A=2; A003462 has A=3; A002450 has A=4; A003463 has A=5; A003464 has A=6; A023000 has A=7; A023001 has A=8; A002452 has A=9; A002275 has A=10; A016123 has A=11; A016125 has A=12; A091030 has A=13; A135519 has A=14; A135518 has A=15; A131865 has A=16; A091045 has A=17; A064108 has A=20. - _Ctibor O. Zizka_, Mar 03 2008
%C a(n) is also a Mersenne prime A000668 when n is a prime number in A000043. - _Omar E. Pol_, Aug 31 2008
%C a(n) is also a Mersenne number A001348 when n is prime. - _Omar E. Pol_, Sep 05 2008
%C With offset 1, = row sums of triangle A144081; and INVERT transform of A009545 starting with offset 1; where A009545 = expansion of sin(x)*exp(x). - _Gary W. Adamson_, Sep 10 2008
%C Numbers n such that A000120(n)/A070939(n) = 1. - _Ctibor O. Zizka_, Oct 15 2008
%C For n > 0, sequence is equal to partial sums of A000079; a(n) = A000203(A000079(n-1)). - _Lekraj Beedassy_, May 02 2009
%C Starting with offset 1 = the Jacobsthal sequence, A001045, (1, 1, 3, 5, 11, 21, ...) convolved with (1, 2, 2, 2, ...). - _Gary W. Adamson_, May 23 2009
%C Numbers n such that n=2*phi(n+1)-1. - _Farideh Firoozbakht_, Jul 23 2009
%C a(n) = (a(n-1)+1)-th odd numbers = A005408(a(n-1)) for n >= 1. - _Jaroslav Krizek_, Sep 11 2009
%C Partial sums of a(n) for n >= 0 are A000295(n+1). Partial sums of a(n) for n >= 1 are A000295(n+1) and A130103(n+1). a(n) = A006127(n) - (n+1). - _Jaroslav Krizek_, Oct 16 2009
%C If n is even a(n) mod 3 = 0. This follows from the congruences 2^(2k) - 1 ~ 2*2*...*2 - 1 ~ 4*4*...*4 - 1 ~ 1*1*...*1 - 1 ~ 0 (mod 3). (Note that 2*2*...*2 has an even number of terms.) - _Washington Bomfim_, Oct 31 2009
%C Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=2,(i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 1, a(n)=det(A). - _Milan Janjic_, Jan 26 2010
%C This is the sequence A(0,1;1,2;2) = A(0,1;3,-2;0) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - _Wolfdieter Lang_, Oct 18 2010
%C a(n) = S(n+1,2), a Stirling number of the second kind. See the example below. - _Dennis P. Walsh_, Mar 29 2011
%C Entries of row a(n) in Pascal's triangle are all odd, while entries of row a(n)-1 have alternating parities of the form odd, even, odd, even, ..., odd.
%C Define the bar operation as an operation on signed permutations that flips the sign of each entry. Then a(n+1) is the number of signed permutations of length 2n that are equal to the bar of their reverse-complements and avoid the set of patterns {(-2,-1), (-1,+2), (+2,+1)}. (See the Hardt and Troyka reference.) - _Justin M. Troyka_, Aug 13 2011
%C A159780(a(n)) = n and A159780(m) < n for m < a(n). - _Reinhard Zumkeller_, Oct 21 2011
%C This sequence is also the number of proper subsets of a set with n elements. - _Mohammad K. Azarian_, Oct 27 2011
%C a(n) is the number k such that the number of iterations of the map k -> (3k +1)/2 == 1 (mod 2) until reaching (3k +1)/2 == 0 (mod 2) equals n. (see the Collatz problem). - _Michel Lagneau_, Jan 18 2012
%C For integers a, b, denote by a<+>b the least c >= a such that Hd(a,c) = b (note that, generally speaking, a<+>b differs from b<+>a). Then a(n+1)=a(n)<+>1. Thus this sequence is the Hamming analog of nonnegative integers. - _Vladimir Shevelev_, Feb 13 2012
%C Pisano period lengths: 1, 1, 2, 1, 4, 2, 3, 1, 6, 4, 10, 2, 12, 3, 4, 1, 8, 6, 18, 4, ... apparently A007733. - _R. J. Mathar_, Aug 10 2012
%C Start with n. Each n generates a sublist {n-1,n-2,...,1}. Each element of each sublist also generates a sublist. Take the sum of all. E.g., 3->{2,1} and 2->{1}, so a(3)=3+2+1+1=7. - _Jon Perry_, Sep 02 2012
%C This is the Lucas U(P=3,Q=2) sequence. - _R. J. Mathar_, Oct 24 2012
%C The Mersenne numbers >= 7 are all Brazilian numbers, as repunits in base two. See Proposition 1 & 5.2 in Links: "Les nombres brésiliens". - _Bernard Schott_, Dec 26 2012
%C Number of line segments after n-th stage in the H tree. - _Omar E. Pol_, Feb 16 2013
%C Row sums of triangle in A162741. - _Reinhard Zumkeller_, Jul 16 2013
%C a(n) is the highest power of 2 such that 2^a(n) divides (2^n)!. - _Ivan N. Ianakiev_, Aug 17 2013
%C In computer programming, these are the only unsigned numbers such that k&(k+1)=0, where & is the bitwise AND operator and numbers are expressed in binary. - _Stanislav Sykora_, Nov 29 2013
%C Minimal number of moves needed to interchange n frogs in the frogs problem (see for example the NRICH 1246 link or the Britton link below). - _N. J. A. Sloane_, Jan 04 2014
%C a(n) !== 4 (mod 5); a(n) !== 10 (mod 11); a(n) !== 2, 4, 5, 6 (mod 7). - _Carmine Suriano_, Apr 06 2014
%C After 0, antidiagonal sums of the array formed by partial sums of integers (1, 2, 3, 4, ...). - _Luciano Ancora_, Apr 24 2015
%C a(n+1) equals the number of ternary words of length n avoiding 01,02. - _Milan Janjic_, Dec 16 2015
%C With offset 0 and another initial 0, the n-th term of 0, 0, 1, 3, 7, 15, ... is the number of commas required in the fully-expanded von Neumann definition of the ordinal number n. For example, 4 := {0, 1, 2, 3} := {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}}, which uses seven commas. Also, for n>0, a(n) is the total number of symbols required in the fully-expanded von Neumann definition of ordinal n - 1, where a single symbol (as usual) is always used to represent the empty set and spaces are ignored. E.g., a(5) = 31, the total such symbols for the ordinal 4. - _Rick L. Shepherd_, May 07 2016
%C With the quantum integers defined by [n+1]_q = (q^(n+1) - q^(-n-1)) / (q - q^(-1)), the Mersenne numbers are a(n+1) = q^n [n+1]_q with q = sqrt(2), whereas the signed Jacobsthal numbers A001045 are given by q = i * sqrt(2) for i^2 = -1. Cf. A239473. - _Tom Copeland_, Sep 05 2016
%C For n>1: numbers n such that n - 1 divides sigma(n + 1). - _Juri-Stepan Gerasimov_, Oct 08 2016
%C This is also the second column of the Stirling2 triangle A008277 (see also A048993). - _Wolfdieter Lang_, Feb 21 2017
%C Except for the initial terms, the decimal representation of the x-axis of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 659", "Rule 721" and "Rule 734", based on the 5-celled von Neumann neighborhood initialized with a single on cell. - _Robert Price_, Mar 14 2017
%C a(n), n > 1, is the number of maximal subsemigroups of the monoid of order-preserving partial injective mappings on a set with n elements. - _James Mitchell_ and _Wilf A. Wilson_, Jul 21 2017
%C Also the number of independent vertex sets and vertex covers in the complete bipartite graph K_{n-1,n-1}. - _Eric W. Weisstein_, Sep 21 2017
%C Sum_{k=0..n} p^k is the determinant of n X n matrix M_(i, j) = binomial(i + j - 1, j)*p + binomial(i+j-1, i), in this case p=2 (empirical observation). - _Tony Foster III_, May 11 2019
%C The rational numbers r(n) = a(n+1)/2^(n+1) = a(n+1)/A000079(n+1) appear also as root of the n-th iteration f^{[n]}(c; x) = 2^(n+1)*x - a(n+1)*c of f(c; x) = f^{[0]}(c; x) = 2*x - c as r(n)*c. This entry is motivated by a riddle of Johann Peter Hebel (1760 - 1826): Erstes Rechnungsexempel(Ein merkwürdiges Rechnungs-Exempel) from 1803, with c = 24 and n = 2, leading to the root r(2)*24 = 21 as solution. See the link and reference. For the second problem, also involving the present sequence, see a comment in A130330. - _Wolfdieter Lang_, Oct 28 2019
%C a(n) is the sum of the smallest elements of all subsets of {1,2,..,n} that contain n. For example, a(3)=7; the subsets of {1,2,3} that contain 3 are {3}, {1,3}, {2,3}, {1,2,3}, and the sum of smallest elements is 7. - _Enrique Navarrete_, Aug 21 2020
%C a(n-1) is the number of nonempty subsets of {1,2,..,n} which don't have an element that is the size of the set. For example, for n = 4, a(3) = 7 and the subsets are {2}, {3}, {4}, {1,3}, {1,4}, {3,4}, {1,2,4}. - _Enrique Navarrete_, Nov 21 2020
%C From _Eric W. Weisstein_, Sep 04 2021: (Start)
%C Also the number of dominating sets in the complete graph K_n.
%C Also the number of minimum dominating sets in the n-helm graph for n >= 3. (End)
%C Conjecture: except for a(2)=3, numbers m such that 2^(m+1) - 2^j - 2^k - 1 is composite for all 0 <= j < k <= m. - _Chai Wah Wu_, Sep 08 2021
%C a(n) is the number of three-in-a-rows passing through a corner cell in n-dimensional tic-tac-toe. - _Ben Orlin_, Mar 15 2022
%C From _Vladimir Pletser_, Jan 27 2023: (Start)
%C a(n) == 1 (mod 30) for n == 1 (mod 4);
%C a(n) == 7 (mod 120) for n == 3 (mod 4);
%C (a(n) - 1)/30 = (a(n+2) - 7)/120 for n odd;
%C (a(n) - 1)/30 = (a(n+2) - 7)/120 = A131865(m) for n == 1 (mod 4) and m >= 0 with A131865(0) = 0. (End)
%C a(n) is the number of n-digit numbers whose smallest decimal digit is 8. - _Stefano Spezia_, Nov 15 2023
%C Also, number of nodes in a perfect binary tree of height n-1, or: number of squares (or triangles) after the n-th step of the construction of a Pythagorean tree: Start with a segment. At each step, construct squares having the most recent segment(s) as base, and isoceles right triangles having the opposite side of the squares as hypotenuse ("on top" of each square). The legs of these triangles will serve as the segments which are the bases of the squares in the next step. - _M. F. Hasler_, Mar 11 2024
%C a(n) is the length of the longest path in the n-dimensional hypercube. - _Christian Barrientos_, Apr 13 2024
%D P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 75.
%D Ralph P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, Addison-Wesley, 2004, p. 134.
%D Johann Peter Hebel, Gesammelte Werke in sechs Bänden, Herausgeber: Jan Knopf, Franz Littmann und Hansgeorg Schmidt-Bergmann unter Mitarbeit von Ester Stern, Wallstein Verlag, 2019. Band 3, S. 20-21, Loesung, S. 36-37. See also the link below.
%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 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, "Tower of Hanoi", Penguin Books, 1987, pp. 112-113.
%H Franklin T. Adams-Watters, <a href="/A000225/b000225.txt">Table of n, a(n) for n = 0..1000</a>
%H Omran Ahmadi and Robert Granger, <a href="https://doi.org/10.1090/S0025-5718-2013-02705-6">An efficient deterministic test for Kloosterman sum zeros</a>, Math. Comp. 83 (2014), 347-363, arXiv:<a href="https://arxiv.org/abs/1104.3882">1104.3882</a> [math.NT], 2011-2012. See 1st and 2nd column of Table 1 p. 9.
%H Feryal Alayont and Evan Henning, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Alayont/ala4.html">Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4.
%H Anonymous, <a href="http://www.geider.net/eng/math/hanoi.htm">The Tower of Hanoi</a>
%H M. Baake, F. Gahler and U. Grimm, <a href="https://arxiv.org/abs/1211.5466">Examples of substitution systems and their factors</a>, arXiv:1211.5466 [math.DS], 2012-2013.
%H Michael Baake, Franz Gähler, and Uwe Grimm, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Baake/baake3.html">Examples of Substitution Systems and Their Factors</a>, Journal of Integer Sequences, Vol. 16 (2013), #13.2.14.
%H J.-L. Baril, <a href="https://doi.org/10.37236/665">Classical sequences revisited with permutations avoiding dotted pattern</a>, Electronic Journal of Combinatorics, 18 (2011), #P178.
%H Paul Barry, <a href="https://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 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 J. Bernheiden, <a href="https://web.archive.org/web/20130314031122/http://www.mathe-schule.de/download/pdf/Primzahl/Mersenne.pdf">Mersennesche Zahlen</a>, (Text in German) [Wayback Machine cached version].
%H Michael Boardman, <a href="http://www.jstor.org/stable/3219201">The Egg-Drop Numbers</a>, Mathematics Magazine, 77 (2004), 368-372.
%H R. P. Brent and H. J. J. te Riele, <a href="https://web.archive.org/web/20190216110823/https://ir.cwi.nl/pub/5423/05423D.pdf">Factorizations of a^n +- 1, 13 <= a < 100</a>, CWI Report 9212, 1992 [Wayback Machine cached version].
%H R. P. Brent, P. L. Montgomery and H. J. J. te Riele, <a href="https://pdfs.semanticscholar.org/1735/97b016f4c239adaabf9f8afda341959fa422.pdf">Factorizations of a^n +- 1, 13 <= a < 100: Update 2</a>
%H R. P. Brent, P. L. Montgomery and H. J. J. te Riele, <a href="http://citeseer.ifi.unizh.ch/correct/467051">Factorizations Of Cunningham Numbers With Bases 13 To 99. Millennium Edition</a> [BROKEN LINK]
%H R. P. Brent, P. L. Montgomery and H. J. J. te Riele, <a href="http://wwwmaths.anu.edu.au/~brent/pub/pub200.html">Factorizations of Cunningham numbers with bases 13 to 99: Millennium edition</a>
%H R. P. Brent and H. J. J. te Riele, <a href="http://wwwmaths.anu.edu.au/~brent/pub/pub134.html">Factorizations of a^n +- 1, 13 <= a < 100</a>
%H John Brillhart et al., <a href="https://doi.org/10.1090/conm/022">Cunningham Project</a> [Factorizations of b^n +- 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers] [Subscription required].
%H Jill Britton, <a href="https://web.archive.org/web/20160426145336/http://britton.disted.camosun.bc.ca/hanoi.swf">The Tower of Hanoi</a> [Video file, Wayback Machine cached version].
%H Jill Britton, <a href="https://web.archive.org/web/20170608141000/http://britton.disted.camosun.bc.ca:80/frog_puzzle.htm">The Frog Puzzle</a> [Wayback Machine cached version].
%H C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=MersenneNumber">Mersenne number</a>
%H Naiomi T. Cameron and Asamoah Nkwanta, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Cameron/cameron46.html">On Some (Pseudo) Involutions in the Riordan Group</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
%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. Catarino, H. Campos, and P. Vasco, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_46_from37to53.pdf">On the Mersenne sequence</a>, Annales Mathematicae et Informaticae, 46 (2016) pp. 37-53.
%H Robert George Cowell, <a href="https://arxiv.org/abs/1802.09863">A unifying framework for the modelling and analysis of STR DNA samples arising in forensic casework</a>, arXiv:1802.09863 [stat.AP], 2018.
%H F. Javier de Vega, <a href="https://arxiv.org/abs/2003.13378">An extension of Furstenberg's theorem of the infinitude of primes</a>, arXiv:2003.13378 [math.NT], 2020.
%H W. M. B. Dukes, <a href="https://arxiv.org/abs/math/0411557">On the number of matroids on a finite set</a>, arXiv:math/0411557 [math.CO], 2004.
%H James East, Jitender Kumar, James D. Mitchell, and Wilf A. Wilson, <a href="https://arxiv.org/abs/1706.04967">Maximal subsemigroups of finite transformation and partition monoids</a>, arXiv:1706.04967 [math.GR], 2017. [_James Mitchell_ and _Wilf A. Wilson_, Jul 21 2017]
%H W. Edgington, <a href="http://www.garlic.com/~wedgingt/mersenne.htm">Mersenne Page</a> [BROKEN LINK]
%H David Eppstein, <a href="https://arxiv.org/abs/1804.07396">Making Change in 2048</a>, arXiv:1804.07396 [cs.DM], 2018.
%H T. Eveilleau, <a href="https://mathsmagiques.fr/pages/jeux_mat/textes/hanoi.html">Animated solution to the Tower of Hanoi problem</a>
%H G. Everest et al., <a href="http://www.jstor.org/stable/27642221">Primes generated by recurrence sequences</a>, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
%H G. Everest, S. Stevens, D. Tamsett and T. Ward, <a href="https://arxiv.org/abs/math/0412079">Primitive divisors of quadratic polynomial sequences</a>, arXiv:math/0412079 [math.NT], 2004-2006.
%H G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Ward/ward2.html">Integer Sequences and Periodic Points</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3.
%H Bakir Farhi, <a href="https://www.emis.de/journals/JIS/VOL22/Farhi/farhi19.html">Summation of Certain Infinite Lucas-Related Series</a>, J. Int. Seq., Vol. 22 (2019), Article 19.1.6.
%H Emmanuel Ferrand, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Ferrand/ferrand8.html">Deformations of the Taylor Formula</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.7.
%H Robert Frontczak and Taras Goy, <a href="https://journals.pnu.edu.ua/index.php/cmp/article/view/3870">Mersenne-Horadam identities using generating functions</a>, Carpathian Mathematical Publications, Vol. 12, no. 1, (2020), 34-45.
%H Robert Granger, <a href="https://arxiv.org/abs/1610.06878">On the Enumeration of Irreducible Polynomials over GF(q) with Prescribed Coefficients</a>, arXiv:1610.06878 [math.AG], 2016. See 1st and 2nd column of Table 1 p. 13.
%H Taras Goy, <a href="http://www.math.nthu.edu.tw/~amen/">On new identities for Mersenne numbers</a>, Applied Mathematics E-Notes, 18 (2018), 100-105.
%H R. K. Guy, <a href="/A000346/a000346.pdf">Letter to N. J. A. Sloane</a>
%H A. Hardt and J. M. Troyka, <a href="http://www.mat.unisi.it/newsito/puma/public_html/23_3/hardt_troyka.pdf">Restricted symmetric signed permutations</a>, Pure Mathematics and Applications, Vol. 23 (No. 3, 2012), pp. 179--217.
%H A. Hardt and J. M. Troyka, <a href="https://apps.carleton.edu/curricular/math/assets/Andy_hardt_slides.pdf">Slides</a> (associated with the Hardt and Troyka reference above).
%H Johann Peter Hebel, <a href="https://gutenberg.spiegel.de/buch/schatzkastlein-des-rheinischen-hausfreundes-8818/6">Erstes Rechnungsexempel</a>, 1803; Solution: <a href="https://gutenberg.spiegel.de/buch/schatzkastlein-des-rheinischen-hausfreundes-8818/15">Auflösung des ersten Rechnungsexempels und ein zweites</a>, 1804.
%H A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, <a href="http://dx.doi.org/10.1007/978-3-0348-0237-6">The Tower of Hanoi - Myths and Maths</a>, Birkhäuser 2013. See page 11. <a href="http://tohbook.info">Book's website</a>
%H Andreas M. Hinz and Paul K. Stockmeyer, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Hinz/hinz5.html">Precious Metal Sequences and Sierpinski-Type Graphs</a>, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=138">Encyclopedia of Combinatorial Structures 138</a>, <a href="http://ecs.inria.fr/services/structure?nbr=345">345</a>, <a href="http://ecs.inria.fr/services/structure?nbr=371">371</a>, and <a href="http://ecs.inria.fr/services/structure?nbr=880">880</a>
%H Jiří Klaška, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Klaska/klaska6.html">Jakóbczyk's Hypothesis on Mersenne Numbers and Generalizations of Skula's Theorem</a>, J. Int. Seq., Vol. 26 (2023), Article 23.3.8.
%H Ross La Haye, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/LaHaye/lahaye5.html">Binary Relations on the Power Set of an n-Element Set</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
%H Wolfdieter Lang, <a href="/A000225/a000225.pdf">Notes on certain inhomogeneous three term recurrences.</a>
%H J. Loy, <a href="http://www.jimloy.com/puzz/hanoi.htm">The Tower of Hanoi</a>
%H Edouard Lucas, <a href="http://www.mathstat.dal.ca/FQ/Books/Complete/simply-periodic.pdf">The Theory of Simply Periodic Numerical Functions</a>, Fibonacci Association, 1969. English translation of article "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240.
%H Mathforum, <a href="http://mathforum.org/dr.math/faq/faq.tower.hanoi.html">Tower of Hanoi</a>
%H Mathforum, Problem of the Week, <a href="http://mathforum.org/midpow/solutions/solution.ehtml?puzzle=17">The Tower of Hanoi Puzzle</a>
%H Donatella Merlini and Massimo Nocentini, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Merlini/merlini5.html">Algebraic Generating Functions for Languages Avoiding Riordan Patterns</a>, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.3.
%H R. Mestrovic, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - From _N. J. A. Sloane_, Jun 13 2012
%H N. Moreira and R. Reis, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Moreira/moreira8.html">On the Density of Languages Representing Finite Set Partitions</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
%H N. Neumarker, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/">Realizability of Integer Sequences as Differences of Fixed Point Count Sequences< a>, JIS 12 (2009) 09.4.5, Example 11.
%H NRICH 1246, <a href="http://nrich.maths.org/1246">Frogs</a>
%H Ahmet Öteleş, <a href="http://www.kurims.kyoto-u.ac.jp/EMIS/journals/ASUO/mathematics/anale2019vol2/06_oteles.pdf">Bipartite Graphs Associated with Pell, Mersenne and Perrin Numbers</a>, An. Şt. Univ. Ovidius Constantą, (2019) Vol. 27, Issue 2, 109-120.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H Bernard Schott, <a href="/A125134/a125134.pdf">Les nombres brésiliens</a>, Reprinted from Quadrature, no. 76, avril-juin 2010, pages 30-38, included here with permission from the editors of Quadrature.
%H R. R. Snapp, <a href="http://www.cs.uvm.edu/~snapp/teaching/CS5/lectures/hanoi.pdf">The Tower of Hanoi</a>
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.2634312">On the generalized sums of Mersenne, Fermat, Cullen and Woodall Numbers</a>, Politecnico di Torino (Italy, 2019).
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.18483/ijSci.2044">Composition Operations of Generalized Entropies Applied to the Study of Numbers</a>, International Journal of Sciences (2019) Vol. 8, No. 4, 87-92.
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3471358">The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences</a>, Politecnico di Torino, Italy (2019), [math.NT].
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.18483/ijSci.2188">Some Groupoids and their Representations by Means of Integer Sequences</a>, International Journal of Sciences (2019) Vol. 8, No. 10.
%H Thesaurus.maths.org, <a href="http://thesaurus.maths.org/dictionary/map/word/3371">Mersenne Number</a>
%H Thinks.com, <a href="http://thinks.com/java/hanoi/hanoi.htm">Tower of Hanoi, A classic puzzle game</a>
%H A. Umar, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Umar/umar2.html">Combinatorial Results for Semigroups of Orientation-Preserving Partial Transformations</a>, Journal of Integer Sequences, 14 (2011), #11.7.5.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CoinTossing.html">Coin Tossing</a>, <a href="http://mathworld.wolfram.com/Digit.html">Digit</a>, <a href="http://mathworld.wolfram.com/Repunit.html">Repunit</a>, <a href="http://mathworld.wolfram.com/Rule222.html">Rule 222</a>, <a href="http://mathworld.wolfram.com/Run.html">Run</a>, and <a href="http://mathworld.wolfram.com/TowerofHanoi.html">Tower of Hanoi</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteGraph.html">Complete
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DominatingSet.html">Dominating Set</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HelmGraph.html">Helm Graph</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="https://mathworld.wolfram.com/MersenneNumber.html">Mersenne Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MinimumDominatingSet.html">Minimum Dominating Set</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/H_tree">H tree</a>, <a href="http://en.wikipedia.org/wiki/Lucas_sequence">Lucas sequence</a>, and <a href="http://en.wikipedia.org/wiki/Tower_of_Hanoi">Tower of Hanoi</a>
%H K. K. Wong, <a href="http://www.lhs.berkeley.edu/Java/Tower/Tower.html">Tower Of Hanoi:Online Game</a>
%H K. Zsigmondy, <a href="https://doi.org/10.1007%2FBF01692444">Zur Theorie der Potenzreste</a>, Monatsh. Math., 3 (1892), 265-284.
%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/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).
%F G.f.: x/((1-2*x)*(1-x)).
%F E.g.f.: exp(2*x) - exp(x).
%F E.g.f. if offset 1: ((exp(x)-1)^2)/2.
%F a(n) = Sum_{k=0..n-1} 2^k. - _Paul Barry_, May 26 2003
%F a(n) = a(n-1) + 2*a(n-2) + 2, a(0)=0, a(1)=1. - _Paul Barry_, Jun 06 2003
%F Let b(n) = (-1)^(n-1)*a(n). Then b(n) = Sum_{i=1..n} i!*i*Stirling2(n,i)*(-1)^(i-1). E.g.f. of b(n): (exp(x)-1)/exp(2x). - Mario Catalani (mario.catalani(AT)unito.it), Dec 19 2003
%F a(n+1) = 2*a(n) + 1, a(0) = 0.
%F a(n) = Sum_{k=1..n} binomial(n, k).
%F a(n) = n + Sum_{i=0..n-1} a(i); a(0) = 0. - _Rick L. Shepherd_, Aug 04 2004
%F a(n+1) = (n+1)*Sum_{k=0..n} binomial(n, k)/(k+1). - _Paul Barry_, Aug 06 2004
%F a(n+1) = Sum_{k=0..n} binomial(n+1, k+1). - _Paul Barry_, Aug 23 2004
%F Inverse binomial transform of A001047. Also U sequence of Lucas sequence L(3, 2). - _Ross La Haye_, Feb 07 2005
%F a(n) = A099393(n-1) - A020522(n-1) for n > 0. - _Reinhard Zumkeller_, Feb 07 2006
%F a(n) = A119258(n,n-1) for n > 0. - _Reinhard Zumkeller_, May 11 2006
%F a(n) = 3*a(n-1) - 2*a(n-2); a(0)=0, a(1)=1. - _Lekraj Beedassy_, Jun 07 2006
%F Sum_{n>0} 1/a(n) = 1.606695152... = A065442, see A038631. - _Philippe Deléham_, Jun 27 2006
%F Stirling_2(n-k,2) starting from n=k+1. - _Artur Jasinski_, Nov 18 2006
%F a(n) = A125118(n,1) for n > 0. - _Reinhard Zumkeller_, Nov 21 2006
%F a(n) = StirlingS2(n+1,2). - _Ross La Haye_, Jan 10 2008
%F a(n) = A024036(n)/A000051(n). - _Reinhard Zumkeller_, Feb 14 2009
%F a(n) = A024088(n)/A001576(n). -_Reinhard Zumkeller_, Feb 15 2009
%F a(2*n) = a(n)*A000051(n); a(n) = A173787(n,0). - _Reinhard Zumkeller_, Feb 28 2010
%F For n > 0: A179857(a(n)) = A024036(n) and A179857(m) < A024036(n) for m < a(n). - _Reinhard Zumkeller_, Jul 31 2010
%F From _Enrique Pérez Herrero_, Aug 21 2010: (Start)
%F a(n) = J_n(2), where J_n is the n-th Jordan Totient function: (A007434, is J_2).
%F a(n) = Sum_{d|2} d^n*mu(2/d). (End)
%F A036987(a(n)) = 1. - _Reinhard Zumkeller_, Mar 06 2012
%F a(n+1) = A044432(n) + A182028(n). - _Reinhard Zumkeller_, Apr 07 2012
%F a(n) = A007283(n)/3 - 1. - _Martin Ettl_, Nov 11 2012
%F a(n+1) = A001317(n) + A219843(n); A219843(a(n)) = 0. - _Reinhard Zumkeller_, Nov 30 2012
%F a(n) = det(|s(i+2,j+1)|, 1 <= i,j <= n-1), where s(n,k) are Stirling numbers of the first kind. - _Mircea Merca_, Apr 06 2013
%F G.f.: Q(0), where Q(k) = 1 - 1/(4^k - 2*x*16^k/(2*x*4^k - 1/(1 - 1/(2*4^k - 8*x*16^k/(4*x*4^k - 1/Q(k+1)))))); (continued fraction). - _Sergei N. Gladkovskii_, May 22 2013
%F E.g.f.: Q(0), where Q(k) = 1 - 1/(2^k - 2*x*4^k/(2*x*2^k - (k+1)/Q(k+1))); (continued fraction).
%F G.f.: Q(0), where Q(k) = 1 - 1/(2^k - 2*x*4^k/(2*x*2^k - 1/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 23 2013
%F a(n) = A000203(2^(n-1)), n >= 1. - _Ivan N. Ianakiev_, Aug 17 2013
%F a(n) = Sum_{t_1+2*t_2+...+n*t_n=n} n*multinomial(t_1+t_2 +...+t_n,t_1,t_2,...,t_n)/(t_1+t_2 +...+t_n). - _Mircea Merca_, Dec 06 2013
%F a(0) = 0; a(n) = a(n-1) + 2^(n-1) for n >= 1. - _Fred Daniel Kline_, Feb 09 2014
%F a(n) = A125128(n) - A000325(n) + 1. - _Miquel Cerda_, Aug 07 2016
%F From _Ilya Gutkovskiy_, Aug 07 2016: (Start)
%F Binomial transform of A057427.
%F Sum_{n>=0} a(n)/n! = A090142. (End)
%F a(n) = A000918(n) + 1. - _Miquel Cerda_, Aug 09 2016
%F a(n+1) = (A095151(n+1) - A125128(n))/2. - _Miquel Cerda_, Aug 12 2016
%F a(n) = (A079583(n) - A000325(n+1))/2. - _Miquel Cerda_, Aug 15 2016
%F Convolution of binomial coefficient C(n,a(k)) with itself is C(n,a(k+1)) for all k >= 3. - _Anton Zakharov_, Sep 05 2016
%F a(n) = (A083706(n-1) + A000325(n))/2. - _Miquel Cerda_, Sep 30 2016
%F a(n) = A005803(n) + A005408(n-1). - _Miquel Cerda_, Nov 25 2016
%F a(n) = A279396(n+2,2). - _Wolfdieter Lang_, Jan 10 2017
%F a(n) = n + Sum_{j=1..n-1} (n-j)*2^(j-1). See a Jun 14 2017 formula for A000918(n+1) with an interpretation. - _Wolfdieter Lang_, Jun 14 2017
%F a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} C(k,i). - _Wesley Ivan Hurt_, Sep 21 2017
%F a(n+m) = a(n)*a(m) + a(n) + a(m). - _Yuchun Ji_, Jul 27 2018
%F a(n+m) = a(n+1)*a(m) - 2*a(n)*a(m-1). - _Taras Goy_, Dec 23 2018
%F a(n+1) is the determinant of n X n matrix M_(i, j) = binomial(i + j - 1, j)*2 + binomial(i+j-1, i) (empirical observation). - _Tony Foster III_, May 11 2019
%e For n=3, a(3)=S(4,2)=7, a Stirling number of the second kind, since there are 7 ways to partition {a,b,c,d} into 2 nonempty subsets, namely,
%e {a}U{b,c,d}, {b}U{a,c,d}, {c}U{a,b,d}, {d}U{a,b,c}, {a,b}U{c,d}, {a,c}U{b,d}, and {a,d}U{b,c}. - _Dennis P. Walsh_, Mar 29 2011
%e From _Justin M. Troyka_, Aug 13 2011: (Start)
%e Since a(3) = 7, there are 7 signed permutations of 4 that are equal to the bar of their reverse-complements and avoid {(-2,-1), (-1,+2), (+2,+1)}. These are:
%e (+1,+2,-3,-4),
%e (+1,+3,-2,-4),
%e (+1,-3,+2,-4),
%e (+2,+4,-1,-3),
%e (+3,+4,-1,-2),
%e (-3,+1,-4,+2),
%e (-3,-4,+1,+2). (End)
%e G.f. = x + 3*x^2 + 7*x^3 + 15*x^4 + 31*x^5 + 63*x^6 + 127*x^7 + ...
%p A000225 := n->2^n-1; [ seq(2^n-1,n=0..50) ];
%p A000225:=1/(2*z-1)/(z-1); # _Simon Plouffe_ in his 1992 dissertation, sequence starting at a(1)
%t a[n_] := 2^n - 1; Table[a[n], {n, 0, 30}] (* _Stefan Steinerberger_, Mar 30 2006 *)
%t Array[2^# - 1 &, 50, 0] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006 *)
%t NestList[2 # + 1 &, 0, 32] (* _Robert G. Wilson v_, Feb 28 2011 *)
%t 2^Range[0, 20] - 1 (* _Eric W. Weisstein_, Jul 17 2017 *)
%t LinearRecurrence[{3, -2}, {1, 3}, 20] (* _Eric W. Weisstein_, Sep 21 2017 *)
%t CoefficientList[Series[1/(1 - 3 x + 2 x^2), {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 21 2017 *)
%o (PARI) A000225(n) = 2^n-1 \\ _Michael B. Porter_, Oct 27 2009
%o (Haskell)
%o a000225 = (subtract 1) . (2 ^)
%o a000225_list = iterate ((+ 1) . (* 2)) 0
%o -- _Reinhard Zumkeller_, Mar 20 2012
%o (PARI) concat(0, Vec(x/((1-2*x)*(1-x)) + O(x^100))) \\ _Altug Alkan_, Oct 28 2015
%o (SageMath)
%o def isMersenne(n): return n == sum([(1 - b) << s for (s, b) in enumerate((n+1).bits())]) # _Peter Luschny_, Sep 01 2019
%o (Python)
%o def A000225(n): return (1<<n)-1 # _Chai Wah Wu_, Jul 06 2022
%Y Cf. A000043 (Mersenne exponents).
%Y Cf. A000668 (Mersenne primes).
%Y Cf. A001348 (Mersenne numbers with n prime).
%Y Cf. A000079, A001045, A009545, A016189, A052955, A083329, A085104, A144081.
%Y Cf. a(n)=A112492(n, 2). Rightmost column of A008969.
%Y a(n) = A118654(n, 1) = A118654(n-1, 3), for n > 0.
%Y Subsequence of A132781.
%Y Smallest number whose base b sum of digits is n: this sequence (b=2), A062318 (b=3), A180516 (b=4), A181287 (b=5), A181288 (b=6), A181303 (b=7), A165804 (b=8), A140576 (b=9), A051885 (b=10).
%Y Cf. A000203, A239473, A279396.
%Y Cf. A008277, A048993 (columns k=2), A000918, A130330.
%K nonn,easy,core,nice,changed
%O 0,3
%A _N. J. A. Sloane_
%E Name partially edited by _Eric W. Weisstein_, Sep 04 2021
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