

A000225


a(n) = 2^n  1. (Sometimes called Mersenne numbers, although that name is usually reserved for A001348.)
(Formerly M2655 N1059)


902



0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 16777215, 33554431, 67108863, 134217727, 268435455, 536870911, 1073741823, 2147483647, 4294967295
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OFFSET

0,3


COMMENTS

This is the Gaussian binomial coefficient [n,1] for q=2.
Number of rank1 matroids over S_n.
Numbers n such that central binomial coefficient is odd: A001405(A000225(n)) mod 2 = 1.  Labos Elemer, Mar 12 2003
This gives the (zerobased) positions of odd terms in the following convolution sequences: A000108, A007460, A007461, A007463, A007464, A061922.
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
a(0) = 0, a(1) = 1; a(n) = smallest number such that a(n)a(m) == 0 (mod (nm+1)), for all m.  Amarnath Murthy, Oct 23 2003
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
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
a(n) = A099393(n1)  A020522(n1) for n > 0.  Reinhard Zumkeller, Feb 07 2006
Numbers n for which the expression 2^n/(n+1) is an integer.  Paolo P. Lava, May 12 2006
Number of nonempty subsets of a set with n elements.  Michael Somos, Sep 03 2006
For n >= 2, a(n) is the least Fibonacci nstep number that is not a power of 2.  Rick L. Shepherd, Nov 19 2007
Let P(A) be the power set of an nelement 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
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. AdamsWatters, Oct 28 2011
2^n1 is the sum of the elements in a Pascal triangle of depth n.  Brian Lewis (bsl04(AT)uark.edu), Feb 26 2008
Sequence generalized : a(n)=(A^n 1)/(A1), 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
a(n) is also a Mersenne prime A000668 when n is a prime number in A000043.  Omar E. Pol, Aug 31 2008
a(n) is also a Mersenne number A001348 when n is prime.  Omar E. Pol, Sep 05 2008
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
Numbers n such that A000120(n)/A070939(n) = 1.  Ctibor O. Zizka, Oct 15 2008
a(n) = A024036(n)/A000051(n).  Reinhard Zumkeller, Feb 14 2009
For n > 0, sequence is equal to partial sums of A000079 ; a(n) = A000203(A000079(n1)).  Lekraj Beedassy, May 02 2009
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
Numbers n such that n=2*phi(n+1)1.  Farideh Firoozbakht, Jul 23 2009
a(n) = (a(n1)+1) th odd numbers = A005408(a(n1)) for n >= 1.  Jaroslav Krizek, Sep 11 2009
a(n) = sum of previous terms + n = (Sum_{i=0..n1} a(i)) + n for n >= 1. 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
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
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=2,(i>1), A[i,i1]=1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A).  Milan Janjic, Jan 26 2010
a(2*n) = a(n)*A000051(n); a(n) = A173787(n,0).  Reinhard Zumkeller, Feb 28 2010
For n>0: A179857(a(n))=A024036(n) and A179857(m)<A024036(n) for m<a(n).  Reinhard Zumkeller, Jul 31 2010
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
a(n)=S(n+1,2), a Stirling number of the second kind. See the example below.  Dennis P. Walsh, Mar 29 2011
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.
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 reversecomplements and avoid the set of patterns {(2,1), (1,+2), (+2,+1)}. (See the Hardt and Troyka reference.)  Justin M. Troyka, Aug 13 2011
A159780(a(n)) = n and A159780(m) < n for m < a(n).  Reinhard Zumkeller, Oct 21 2011
This sequence is also the number of proper subsets of a set with n elements.  Mohammad K. Azarian, Oct 27 2011
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
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
A036987(a(n)) = 1.  Reinhard Zumkeller, Mar 06 2012
a(n+1) = A044432(n) + A182028(n).  Reinhard Zumkeller, Apr 07 2012
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
Start with n. Each n generates a sublist {n1,n2,..,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
This is the Lucas U(P=3,Q=2) sequence.  R. J. Mathar, Oct 24 2012
a(n+1) = A001317(n) + A219843(n); A219843(a(n)) = 0.  Reinhard Zumkeller, Nov 30 2012
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
Number of line segments after nth stage in the H tree.  Omar E. Pol, Feb 16 2013
Row sums of triangle in A162741.  Reinhard Zumkeller, Jul 16 2013
a(n) is the highest power of 2 such that 2^a(n) divides (2^n)!.  Ivan N. Ianakiev, Aug 17 2013
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
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
a(n) != 4 (mod 5); a(n) != 10 (mod 11); a(n) != 2, 4, 5, 6 (mod 7).  Carmine Suriano, Apr 06 2014
After 0, antidiagonal sums of the array formed by partial sums of integers (1, 2, 3, 4, ...).  Luciano Ancora, Apr 24 2015
a(n+1) equals the number of ternary words of length n avoiding 01,02.  Milan Janjic, Dec 16 2015
With offset 0 and another initial 0, the nth term of 0, 0, 1, 3, 7, 15, ... is the number of commas required in the fullyexpanded 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 fullyexpanded 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
With the quantum integers defined by [n+1]_q = (q^(n+1)  q^(n1)) / (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
For n>1: numbers n such that n  1 divides sigma(n + 1).  JuriStepan Gerasimov, Oct 08 2016
This is also the second column of the Stirling2 triangle A008277 (see also A048993).  Wolfdieter Lang, Feb 21 2017
Except for the initial terms, the decimal representation of the xaxis of the nth stage of growth of the twodimensional cellular automaton defined by "Rule 659", "Rule 721" and "Rule 734", based on the 5celled von Neumann neighborhood initialized with a single on cell.  Robert Price, Mar 14 2017
a(n), n > 1, is the number of maximal subsemigroups of the monoid of orderpreserving partial injective mappings on a set with n elements.  James Mitchell and Wilf A. Wilson, Jul 21 2017
Also the number of independent vertex sets and vertex covers in the complete bipartite graph K_{n1,n1}.  Eric W. Weisstein, Sep 21 2017


REFERENCES

P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 75.
Ralph P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, AddisonWesley, 2004, p. 134.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, "Tower of Hanoi", pp. 1123, Penguin Books 1987.
K. Zsigmondy, Zur Theorie der Potenreste, Monatsh. Math., 3 (1892), 265284.


LINKS

Franklin T. AdamsWatters, Table of n, a(n) for n = 0..1000
Omran Ahmadi, Robert Granger, An efficient deterministic test for Kloosterman sum zeros, arXiv:1104.3882 [math.NT], 20112012. See 1st and 2nd column of Table 1 p. 9.
Anonymous, The Tower of Hanoi
M. Baake, F. Gahler and U. Grimm, Examples of substitution systems and their factors, arXiv:1211.5466 [math.DS], 20122013.
Michael Baake, Franz Gähler, and Uwe Grimm, Examples of Substitution Systems and Their Factors, Journal of Integer Sequences, Vol. 16 (2013), #13.2.14.
J.L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
J. Bernheiden, Mersenne Numbers (Text in German)
Michael Boardman, The EggDrop Numbers, Mathematics Magazine, 77 (2004), 368372.
R. P. Brent and H. J. J. te Riele, Factorizations of a^n + 1, 13 <= a < 100, CWI Report 9212, 1992.
R. P. Brent, P. L. Montgomery and H. J. J. te Riele, Factorizations of a^n + 1, 13 <= a < 100: Update 2
R. P. Brent, P. L. Montgomery and H. J. J. te Riele, Factorizations Of Cunningham Numbers With Bases 13 To 99. Millennium Edition
R. P. Brent, P. L. Montgomery and H. J. J. te Riele, Factorizations of Cunningham numbers with bases 13 to 99: Millennium edition
R. P. Brent and H. J. J. te Riele, Factorizations of a^n + 1, 13 <= a < 100
John Brillhart et al., Cunningham Project [Factorizations of b^n + 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers]
Jill Britton, The Tower of Hanoi
Jill Britton, The Frog Puzzle
C. K. Caldwell, The Prime Glossary, Mersenne number
Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
P. Catarino, H. Campos, P. Vasco, On the Mersenne sequence, Annales Mathematicae et Informaticae, 46 (2016) pp. 3753.
W. M. B. Dukes, On the number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.
James East, Jitender Kumar, James D. Mitchell, and Wilf A. Wilson Maximal subsemigroups of finite transformation and partition monoids, arXiv:1706.04967 [math.GR], 2017. [James Mitchell and Wilf A. Wilson, Jul 21 2017]
W. Edgington, Mersenne Page
T. Eveilleau, Animated solution to the Tower of Hanoi problem
G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417431.
G. Everest, S. Stevens, D. Tamsett and T. Ward, Primitive divisors of quadratic polynomial sequences, arXiv:math/0412079 [math.NT], 20042006.
G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, Integer Sequences and Periodic Points, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3
Emmanuel Ferrand, Deformations of the Taylor Formula, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.7.
Robert Granger, On the Enumeration of Irreducible Polynomials over GF(q) with Prescribed Coefficients, arXiv:1610.06878 [math.AG], 2016. See 1st and 2nd column of Table 1 p. 13.
R. K. Guy, Letter to N. J. A. Sloane
A. Hardt and J. M. Troyka, Restricted symmetric signed permutations, Pure Mathematics and Applications, Vol. 23 (No. 3, 2012), pp. 179217.
A. Hardt and J. M. Troyka, Slides (associated with the Hardt and Troyka reference above).
A. M. Hinz, S. Klavžar, U. Milutinović, C. Petr, The Tower of Hanoi  Myths and Maths, Birkhäuser 2013. See page 11. Book's website
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 138, 345, 371, and 880
Ross La Haye, Binary Relations on the Power Set of an nElement Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences.
J. Loy, The Tower of Hanoi
Edouard Lucas, The Theory of Simply Periodic Numerical Functions, Fibonacci Association, 1969. English translation of article "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184240.
Mathforum, Tower of Hanoi
Mathforum, Problem of the Week, The Tower of Hanoi Puzzle
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012  From N. J. A. Sloane, Jun 13 2012
N. Moreira and R. Reis, On the Density of Languages Representing Finite Set Partitions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
N. Neumarker, Realizability of Integer Sequences as Differences of Fixed Point Count Sequences< a>, JIS 12 (2009) 09.4.5, Example 11.
NRICH 1246, Frogs
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Bernard Schott, Les nombres brésiliens, Reprinted from Quadrature, no. 76, avriljuin 2010, pages 3038, included here with permission from the editors of Quadrature.
R. R. Snapp, The Tower of Hanoi
Thesaurus.maths.org, Mersenne Number
Thinks.com, Tower of Hanoi, A classic puzzle game
A. Umar, Combinatorial Results for Semigroups of OrientationPreserving Partial Transformations, Journal of Integer Sequences, 14 (2011), #11.7.5.
Eric Weisstein's World of Mathematics, Coin Tossing, Digit, Mersenne Number, Repunit, Rule 222, Run, and Towers of Hanoi
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Vertex Cover
Wikipedia, H tree, Lucas sequence, and Tower of Hanoi
K. K. Wong, Tower Of Hanoi:Online Game
Index entries for "core" sequences
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (3,2).


FORMULA

G.f.: x/((12*x)*(1x)).
E.g.f.: exp(2*x)exp(x).
E.g.f. if offset 1: ((exp(x)1)^2)/2.
a(n) = Sum_{k=0..n1} 2^k.  Paul Barry, May 26 2003
a(n) = a(n1)+2a(n2)+2, a(0)=0, a(1)=1.  Paul Barry, Jun 06 2003
Let b(n)=(1)^(n1)a(n). Then b(n) = Sum_{i=1..n} i!*i*Stirling2(n,i)*(1)^(i1). E.g.f. of b(n): (exp(x)1)/exp(2x).  Mario Catalani (mario.catalani(AT)unito.it), Dec 19 2003
a(n+1) = 2*a(n) + 1, a(0) = 0.
a(n) = Sum_{k=1..n} binomial(n, k).
a(n) = n + Sum_{i=0..n1} a(i); a(0) = 0.  Rick L. Shepherd, Aug 04 2004
a(n+1) = (n+1)*Sum_{k=0..n} binomial(n, k)/(k+1).  Paul Barry, Aug 06 2004
a(n+1) = Sum_{k=0..n} binomial(n+1, k+1).  Paul Barry, Aug 23 2004
Inverse binomial transform of A001047. Also U sequence of Lucas sequence L(3, 2).  Ross La Haye, Feb 07 2005
a(n) = A119258(n,n1) for n>0.  Reinhard Zumkeller, May 11 2006
a(n) = 3*a(n1)  2*a(n2); a(0)=0,a(1)=1.  Lekraj Beedassy, Jun 07 2006
Sum_{n>0} 1/a(n) = 1.606695152...(ErdősBorwein constant;see A065442, A038631).  Philippe Deléham, Jun 27 2006
Stirling_2(nk,2) starting from n=k+1.  Artur Jasinski, Nov 18 2006
a(n) = A125118(n,1) for n>0.  Reinhard Zumkeller, Nov 21 2006
a(n) = StirlingS2(n+1,2).  Ross La Haye, Jan 10 2008
a(n) = A024088(n)/A001576(n). Reinhard Zumkeller, Feb 15 2009
From Enrique Pérez Herrero, Aug 21 2010: (Start)
a(n) = J_n(2), where J_n is the nth Jordan Totient function: (A007434, is J_2)
a(n) = Sum_{d2} d^n*mu(2/d) (End)
a(n) = (A007283(n)/3)1.  Martin Ettl, Nov 11 2012
a(n) = det(s(i+2,j+1), 1 <= i,j <= n1), where s(n,k) are Stirling numbers of the first kind.  Mircea Merca, Apr 06 2013
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
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).
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
a(n) = A000203(2^(n1)), n>=1.  Ivan N. Ianakiev, Aug 17 2013
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
a(0) = 0; a(n) = a(n1) + 2^(n1) for n>=1.  Fred Daniel Kline, Feb 09 2014
a(n) = A125128(n)  A000325(n) + 1.  Miquel Cerda, Aug 07 2016
From Ilya Gutkovskiy, Aug 07 2016: (Start)
Binomial transform of A057427.
Sum_{n>=0} a(n)/n! = A090142. (End)
a(n) = A000918(n) + 1.  Miquel Cerda, Aug 09 2016
a(n+1) = (A095151(n+1)  A125128(n))/2.  Miquel Cerda, Aug 12 2016
a(n) = (A079583(n)  A000325(n+1))/2.  Miquel Cerda, Aug 15 2016
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
a(n) = (A083706(n1) + A000325(n))/2.  Miquel Cerda, Sep 30 2016
a(n) = A005803(n) + A005408(n1).  Miquel Cerda, Nov 25 2016
a(n) = A279396(n+2,2).  Wolfdieter Lang, Jan 10 2017
a(n) = n + Sum_{j=1..n1} (nj)*2^(j1). See a Jun 14 2017 formula for A000918(n+1) with an interpretation.  Wolfdieter Lang, Jun 14 2017
a(n) = Sum_{k=0..n1} Sum_{i=0..n1} C(k,i).  Wesley Ivan Hurt, Sep 21 2017


EXAMPLE

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,
{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
Since a(3) = 7, there are 7 signed permutations of 4 that are equal to the bar of their reversecomplements and avoid {(2,1), (1,+2), (+2,+1)}. These are:
(+1,+2,3,4),
(+1,+3,2,4),
(+1,3,+2,4),
(+2,+4,1,3),
(+3,+4,1,2),
(3,+1,4,+2),
(3,4,+1,+2).
 Justin M. Troyka, Aug 13 2011
G.f. = x + 3*x^2 + 7*x^3 + 15*x^4 + 31*x^5 + 63*x^6 + 127*x^7 + ...


MAPLE

A000225 := n>2^n1; [ seq(2^n1, n=0..50) ];
A000225:=1/(2*z1)/(z1); # Simon Plouffe in his 1992 dissertation, sequence starting at a(1)


MATHEMATICA

a[n_] := 2^n  1; Table[a[n], {n, 0, 30}] (* Stefan Steinerberger, Mar 30 2006 *)
Array[2^#  1 &, 50, 0] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006 *)
NestList[2 # + 1 &, 0, 32] (* Robert G. Wilson v *)
2^Range[0, 20]  1 (* Eric W. Weisstein, Jul 17 2017 *)
LinearRecurrence[{3, 2}, {1, 3}, 20] (* Eric W. Weisstein, Sep 21 2017 *)
CoefficientList[Series[1/(1  3 x + 2 x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 21 2017 *)


PROG

(PARI) A000225(n) = 2^n1 \\ Michael B. Porter, Oct 27 2009
(Haskell)
a000225 = (subtract 1) . (2 ^)
a000225_list = iterate ((+ 1) . (* 2)) 0
 Reinhard Zumkeller, Mar 20 2012
(PARI) concat(0, Vec(x/((12*x)*(1x)) + O(x^100))) \\ Altug Alkan, Oct 28 2015


CROSSREFS

Cf. A000043, A000079, A000668, A001045, A001348, A009545, A016189, A052955, A083329, A085104, A144081.
Cf. a(n)=A112492(n, 2). Rightmost column of A008969.
a(n) = A118654(n, 1) = A118654(n1, 3), for n > 0.
Subsequence of A132781.
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).
Cf. A000203, A239473, A279396.
Cf. A008277, A048993 (columns k=2), A000918.
Sequence in context: A126646 * A225883 A255047 A168604 A123121 A117060
Adjacent sequences: A000222 A000223 A000224 * A000226 A000227 A000228


KEYWORD

nonn,easy,core,nice


AUTHOR

N. J. A. Sloane


STATUS

approved



