0,3

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 E. (labos(AT)ana.sote.hu), 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 (amarnath_murthy(AT)yahoo.com), 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 (qntmpkt(AT)yahoo.com), 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 (alexandre.wajnberg(AT)ulb.ac.be), Jun 08 2005

a(n) = A099393(n-1) - A020522(n-1) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Feb 07 2006

Numbers n for which the expression 2^n/(n+1) is an integer. - Paolo P. Lava (ppl(AT)spl.at), 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 (rshepherd2(AT)hotmail.com), Nov 19 2007

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 and y are disjoint and for which either x is a subset of y or y is a subset of x. - Ross La Haye (rlahaye(AT)new.rr.com), Jan 10 2008

Also, 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 either 0) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 1) x = y. - Ross La Haye (rlahaye(AT)new.rr.com), Jan 10 2008

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 (ctibor.zizka(AT)seznam.cz), Mar 03 2008

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 75.

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

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.

G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.

Emmanuel Ferrand, Deformations of the Taylor Formula, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.7.

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.

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.

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

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Anonymous, The Tower of Hanoi

J. Bernheiden, Mersenne Numbers (Text in German)

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

R. P. Brent, P. L. Montgomery & H. J. J. te Riele, Factorizations of a^n +- 1, 13 =< a < 100 :Update 2

R. P. Brent, P. L. Montgomery & H. J. J. te Riele, Factorizations Of Cunningham Numbers With Bases 13 To 99. Millennium Edition

R. P. Brent, P. Montgomery & H. te Riele, Factorizations of Cunningham numbers with bases 13 to 99:Millennium edition

R. P. Brent, P. L. Montgomery & H. J. J. te Riele, Factorizations of Cunningham Numbers with Bases 13 to 99:Millennium Edition

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

R. P. Brent, P. L. Montgomery & H. J. J. te Riele, Factorizations of a^n +- 1, 13 =< a < 100. Update 2

John Brillhart et al., Cunningham Project [Factorizations of b^n +- 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers]

J. Britton, The Tower of Hanoi

C. K. Caldwell, The Prime Glossary, Mersenne number

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, 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

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

J. Loy, The Tower of Hanoi

Mathforum, Tower of Hanoi

Mathforum, Problem of the Week, The Tower of Hanoi Puzzle

NationMaster.com, Tower of Hanoi

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

R. R. Snapp, The Tower of Hanoi

Thesaurus.maths.org, Mersenne Number

Thinks.com, Tower of Hanoi, A classic puzzle game

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics

Eric Weisstein's World of Mathematics, Run

Eric Weisstein's World of Mathematics, Rule 222

Wikipedia, Tower of Hanoi

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

Index entries for "core" sequences

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

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

a(n)=a(n-1)+2a(n-2)+2, a(0)=0, a(1)=1. - Paul Barry (pbarry(AT)wit.ie), 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 (rshepherd2(AT)hotmail.com), Aug 04 2004

a(n+1)=(n+1)sum{k=0..n, binomial(n, k)/(k+1)} - Paul Barry (pbarry(AT)wit.ie), Aug 06 2004

a(n+1)=sum{k=0..n, binomial(n+1, k+1)} - Paul Barry (pbarry(AT)wit.ie), Aug 23 2004

Inverse binomial transform of A001047. Also U sequence of Lucas sequence L(3, 2). - Ross La Haye (rlahaye(AT)new.rr.com), Feb 07 2005

a(n) = A119258(n,n-1) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 11 2006

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

Sum_{n=1..inf}1/a(n) = 1,606695152...(Erdos-Borwein constant;see A065442, A038631) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 27 2006

Stirling_2[n-k,2] starting from n=k+1. - Artur Jasinski (grafix(AT)csl.pl), Nov 18 2006

a(n) = A125118(n,1) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Nov 21 2006

a(n) = StirlingS2(n+1,2) - Ross La Haye (rlahaye(AT)new.rr.com), Jan 10 2008

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 (zerinvarylajos(AT)yahoo.com), Dec 01 2006

[seq (stirling2(n, 2) , n=1..33)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo), Dec 06 2006

a:=n->sum (2^j, j=0..n): seq(a(n), n=-1..31); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2007

A000225:=1/(2*z-1)/(z-1); [Conjectured by S. Plouffe in his 1992 dissertation.]

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+1 od: seq(a[n], n=0..32); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 20 2008

a[n_] := 2^n - 1; Table[a[n], {n, 0, 30}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Mar 30 2006

Array[2^# - 1 &, 50, 0] - Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006

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 of A132781.

Adjacent sequences: A000222 A000223 A000224 this_sequence A000226 A000227 A000228

Sequence in context: A097002 A060152 A126646 this_sequence A123121 A117060 A057613

nonn,easy,core,nice

njas

Additional links provided by Lekraj Beedassy (blekraj(AT)yahoo.com), Dec 20 2003