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

 

Logo

Invitation: celebrating 50 years of OEIS, 250000 sequences, and Sloane's 75th, there will be a conference at DIMACS, Rutgers, Oct 9-10 2014.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000225 2^n - 1. (Sometimes called Mersenne numbers, although that name is usually reserved for A001348.)
(Formerly M2655 N1059)
679
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

This is the Gaussian binomial coefficient [n,1] for q=2.

Number of rank-1 matroids over S_n.

Numbers n such that central binomial coefficient is odd : Mod[A001405[A000225(n)],2]=1. - Labos Elemer, Mar 12 2003

This gives the (zero-based) 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 (n-m+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(n-1) - A020522(n-1) 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 n-step number that is not a power of 2. - Rick L. Shepherd, Nov 19 2007

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

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

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

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

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(n-1)). - 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(n-1)+1) th odd numbers = A005408(a(n-1)) for n >= 1. - Jaroslav Krizek, Sep 11 2009

a(n) = sum of previous terms + n = (Sum_(i=0...n-1) 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,i-1]=-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 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

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 {n-1,n-2,..,1}. Each element of each sublist also generates a sublist. Add the lot up. 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 n-th 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

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, Addison-Wesley, 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. 112-3, Penguin Books 1987.

K. Zsigmondy, Zur Theorie der Potenreste, Monatsh. Math., 3 (1892), 265-284.

LINKS

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..1000

Anonymous, The Tower of Hanoi

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 Egg-Drop Numbers, Mathematics Magazine, 77 (2004), 368-372.

R. P. Brent and H. J. J. te Riele, Factorizations of a^n +- 1, 13 =< a < 100

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.

W. M. B. Dukes, On the number of matroids on a finite set

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), 417-431.

G. Everest, S. Stevens, D. Tamsett and T. Ward, Primitive divisors of quadratic polynomial sequences

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.

A. Hardt and J. M. Troyka, Restricted symmetric signed permutations, Pure Mathematics and Applications, Vol. 23 (No. 3, 2012), pp. 179--217.

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 345

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 371

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 880

Ross La Haye, Binary Relations on the Power Set of an n-Element 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

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 BC--2012) and another new proof, Arxiv preprint arXiv:1202.3670, 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.

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, avril-juin 2010, pages 30-38, 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 Orientation-Preserving Partial Transformations, Journal of Integer Sequences, 14 (2011), #11.7.5.

Eric Weisstein's World of Mathematics, Coin Tossing

Eric Weisstein's World of Mathematics, Digit

Eric Weisstein's World of Mathematics, Mersenne Number

Eric Weisstein's World of Mathematics, Repunit

Eric Weisstein's World of Mathematics, Towers of Hanoi

Eric Weisstein's World of Mathematics, Run

Eric Weisstein's World of Mathematics, Rule 222

Wikipedia, H tree

Wikipedia, Lucas sequence

Wikipedia, Tower of Hanoi

K. K. Wong, Tower Of Hanoi:Online Game

Index entries for "core" sequences

Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-2).

Index to divisibility sequences

FORMULA

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

E.g.f.: exp(2*x)-exp(x).

E.g.f. if offset 1: ((exp(x)-1)^2)/2.

a(n)=sum{k=0..n-1, 2^k}. - Paul Barry, May 26 2003

a(n)=a(n-1)+2a(n-2)+2, a(0)=0, a(1)=1. - Paul Barry, Jun 06 2003

Let b(n)=(-1)^(n-1)a(n). Then b(n)=Sum(i!i Stirling2(n, i)(-1)^(i-1), i=1, .., n). 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.

Sum_{k=1..n} C(n, k).

a(n) = n + sum(i=0, n-1, 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,n-1) for n>0. - Reinhard Zumkeller, May 11 2006

a(n) = 3*a(n-1) - 2*a(n-2); a(0)=0,a(1)=1. - Lekraj Beedassy, Jun 07 2006

Sum_{n=1..inf}1/a(n) = 1.606695152...(Erdos-Borwein constant;see A065442, A038631). - Philippe Deléham, Jun 27 2006

Stirling_2[n-k,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 n-th Jordan Totient function: (A007434, is J_2)

a(n) = sum(d|2, 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 <= n-1), 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^(n-1)), 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(n-1) + 2^(n-1) for n>=1. - Fred Daniel Kline, Feb 09 2014

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 reverse-complements 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

MAPLE

A000225 := n->2^n-1; [ seq(2^n-1, n=0..50) ];

seq(add(binomial(n, k)*(bell(k-n)), k=1..n), n=0..32); - Zerinvary Lajos, Dec 01 2006

A000225:=1/(2*z-1)/(z-1); [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 *)

Table[Sum[ Binomial[n + 1, k + 1], {k, 0, n}], {n, -1, 31}] (* Zerinvary Lajos, Jul 08 2009 *)

NestList[2# + 1 &, 0, 32] (* Robert G. Wilson v *)

PROG

(Sage) [gaussian_binomial(n, 1, 2) for n in xrange(1, 33)] # From Zerinvary Lajos, May 24 2009

(PARI) A000225(n) = 2^n-1  \\ Michael B. Porter, Oct 27 2009

(Haskell)

a000225 = (subtract 1) . (2 ^)

a000225_list = iterate ((+ 1) . (* 2)) 0

-- Reinhard Zumkeller, Mar 20 2012

(Maxima)

a[0]:0$ a[1]:1$ a[n]:=a[n-1]+2*a[n-2]+2$ A000225(n):=a[n]$ makelist(A000225(n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */

CROSSREFS

Cf. A000079, A016189.

Cf. a(n)=A112492(n, 2). Rightmost column of A008969.

a(n) = A118654(n, 1) = A118654(n-1, 3), for n > 0.

Subsequence of A132781.

Cf. A000043, A000668, A001348, A009545, A144081, A001045, A052955, A083329.

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. A085104.

Sequence in context: A060152 A126646 * A225883 A168604 A123121 A117060

Adjacent sequences:  A000222 A000223 A000224 * A000226 A000227 A000228

KEYWORD

nonn,easy,core,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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

Content is available under The OEIS End-User License Agreement .

Last modified September 18 05:49 EDT 2014. Contains 246893 sequences.