%I A000225 M2655 N1059
%S A000225 0,1,3,7,15,31,63,127,255,511,1023,2047,4095,8191,16383,32767,65535,
%T A000225 131071,262143,524287,1048575,2097151,4194303,8388607,16777215,33554431,
%U A000225 67108863,134217727,268435455,536870911,1073741823,2147483647,4294967295
%N A000225 2^n - 1. (Sometimes called Mersenne numbers, although that name is usually
reserved for A001348.)
%C A000225 This is the Gaussian binomial coefficient [n,1] for q=2.
%C A000225 Number of rank-1 matroids over S_n.
%C A000225 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
%C A000225 This gives the (zero-based) positions of odd terms in the following convolution
sequences: A000108, A007460, A007461, A007463, A007464, A061922.
%C A000225 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 A000225 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
%C A000225 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
%C A000225 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
%C A000225 a(n) = A099393(n-1) - A020522(n-1) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Feb 07 2006
%C A000225 Numbers n for which the expression 2^n/(n+1) is an integer. - Paolo P.
Lava (ppl(AT)spl.at), May 12 2006
%C A000225 Number of nonempty subsets of a set with n elements. - Michael Somos
Sep 03 2006
%C A000225 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
%C A000225 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
%C A000225 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
%C A000225 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 A000225 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
%C A000225 a(n) is also a Mersenne prime A000668 when n is a prime number A000043.
[From Omar E. Pol (info(AT)polprimos.com), Aug 31 2008]
%C A000225 a(n) is also a Mersenne number A001348 when n is prime. [From Omar E.
Pol (info(AT)polprimos.com), Sep 05 2008]
%C A000225 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).
[From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 10 2008]
%C A000225 Numbers n such that A000120(n)/A070939(n) = 1 [From Ctibor O.Zizka (c.zizka(AT)email.cz),
Oct 15 2008]
%C A000225 a(n) = A024036(n)/A000051(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Feb 14 2009]
%C A000225 For n > 0, sequence is equal to partial sums of A000079 ; a(n) = A000203(A000079(n-1)).
[From Lekraj Beedassy (blekraj(AT)yahoo.com), May 02 2009]
%C A000225 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), May 23 2009:
(Start)
%C A000225 Starting with offset 1 = the Jacobsthal sequence, A001045,
%C A000225 (1, 1, 3, 5, 11, 21,...) convolved with (1, 2, 2, 2,...). (End)
%C A000225 Numbers n such that n=2*phi(n+1)-1. [From Farideh Firoozbakht (mymontain(AT)yahoo.com),
Jul 23 2009]
%C A000225 a(n) = (a(n-1)+1) th odd numbers = A005408(a(n-1)) for n >= 1. [From
Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Sep 11 2009]
%C A000225 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), A125128(n) and A130103(n+1).
a(n) = A006127(n) - (n+1). [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz),
Oct 16 2009]
%C A000225 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.) [From W.
Bomfim (webonfim(AT)bol.com.br), Oct 31 2009]
%D A000225 P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY,
1968, vol. 2, p. 75.
%D A000225 Paul Barry, A Catalan Transform and Related Transformations on Integer
Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%D A000225 Michael Boardman, "The Egg-Drop Numbers", Mathematics Magazine, 77 (2004),
368-372. [From Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Sep
30 2009]
%D A000225 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.
%D A000225 G. Everest et al., Primes generated by recurrence sequences, Amer. Math.
Monthly, 114 (No. 5, 2007), 417-431.
%D A000225 Emmanuel Ferrand, Deformations of the Taylor Formula, Journal of Integer
Sequences, Vol. 10 (2007), Article 07.1.7.
%D A000225 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. [From
Ross La Haye (rlahaye(AT)new.rr.com), Feb 22 2009]
%D A000225 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 A000225 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000225 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000225 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers,
"Tower of Hanoi", pp. 112-3, Penguin Books 1987.
%D A000225 K. Zsigmondy, Zur Theorie der Potenreste, Monatsh. Math., 3 (1892), 265-284.
%H A000225 Franklin T. Adams-Watters, <a href="b000225.txt">Table of n, a(n) for
n = 0..1000</a>
%H A000225 Anonymous, <a href="http://www.geider.net/eng/math/hanoi.htm">The Tower
of Hanoi</a>
%H A000225 J. Bernheiden, <a href="http://www.mathe-schule.de/download/pdf/Primzahl/
Mersenne.pdf">Mersenne Numbers (Text in German)</a>
%H A000225 R. P. Brent and H. J. J. te Riele, <a href="http://ftp.cwi.nl/CWIreports/
1992/NM-R9212.pdf">Factorizations of a^n +- 1, 13 =< a < 100</a>
%H A000225 R. P. Brent, P. L. Montgomery and H. J. J. te Riele, <a href="http://
citeseer.ifi.unizh.ch/correct/32644">Factorizations of a^n +- 1,
13 =< a < 100 :Update 2</a>
%H A000225 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>
%H A000225 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 A000225 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 A000225 John Brillhart et al., <a href="http://www.ams.org/online_bks/conm22">
Cunningham Project</a> [Factorizations of b^n +- 1, b = 2, 3, 5,
6, 7, 10, 11, 12 up to high powers]
%H A000225 J. Britton, <a href="http://britton.disted.camosun.bc.ca/hanoi.swf">The
Tower of Hanoi</a>
%H A000225 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/
page.php?sort=MersenneNumber">Mersenne number</a>
%H A000225 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Sequences realized by oligomorphic permutation groups</a>, J. Integ.
Seqs. Vol. 3 (2000), #00.1.5.
%H A000225 W. M. B. Dukes, <a href="http://arXiv.org/abs/math.CO/0411557">On the
number of matroids on a finite set</a>
%H A000225 W. Edgington, <a href="http://www.garlic.com/~wedgingt/mersenne.htm">
Mersenne Page</a>
%H A000225 T. Eveilleau, <a href="http://perso.orange.fr/therese.eveilleau/pages/
jeux_mat/textes/hanoi.html">Animated solution to the Tower of Hanoi
problem</a>
%H A000225 G. Everest, S. Stevens, D. Tamsett and T. Ward, <a href="http://arXiv.org/
abs/math.NT/0412079">Primitive divisors of quadratic polynomial sequences</
a>
%H A000225 G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, <a href="http:/
/www.cs.uwaterloo.ca/journals/JIS/index.html">Integer Sequences and
Periodic Points</a>, Journal of Integer Sequences, Vol. 5 (2002),
Article 02.2.3
%H A000225 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=138">
Encyclopedia of Combinatorial Structures 138</a>
%H A000225 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=345">
Encyclopedia of Combinatorial Structures 345</a>
%H A000225 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=371">
Encyclopedia of Combinatorial Structures 371</a>
%H A000225 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=880">
Encyclopedia of Combinatorial Structures 880</a>
%H A000225 J. Loy, <a href="http://www.jimloy.com/puzz/hanoi.htm">The Tower of Hanoi</
a>
%H A000225 Mathforum, <a href="http://mathforum.org/dr.math/faq/faq.tower.hanoi.html">
Tower of Hanoi</a>
%H A000225 Mathforum, Problem of the Week, <a href="http://mathforum.org/midpow/
solutions/solution.ehtml?puzzle=17">The Tower of Hanoi Puzzle</a>
%H A000225 NationMaster.com, <a href="http://www.nationmaster.com/encyclopedia/Tower-of-Hanoi">
Tower of Hanoi</a>
%H A000225 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%H A000225 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
1031 Generating Functions and Conjectures</a>, Universit\'{e} du
Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000225 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
index.html">Arithmetic and growth of periodic orbits</a>, J. Integer
Seqs., Vol. 4 (2001), #01.2.1.
%H A000225 R. R. Snapp, <a href="http://www.cs.uvm.edu/~snapp/teaching/CS5/lectures/
hanoi.pdf">The Tower of Hanoi</a>
%H A000225 Thesaurus.maths.org, <a href="http://thesaurus.maths.org/dictionary/map/
word/3371">Mersenne Number</a>
%H A000225 Thinks.com, <a href="http://thinks.com/java/hanoi/hanoi.htm">Tower of
Hanoi, A classic puzzle game</a>
%H A000225 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
CoinTossing.html">Link to a section of The World of Mathematics.</
a>
%H A000225 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Digit.html">Link to a section of The World of Mathematics.</a>
%H A000225 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
MersenneNumber.html">Link to a section of The World of Mathematics.</
a>
%H A000225 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Repunit.html">Link to a section of The World of Mathematics.</a>
%H A000225 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
TowersofHanoi.html">Link to a section of The World of Mathematics</
a>
%H A000225 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Run.html">Run</a>
%H A000225 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Rule222.html">Rule 222</a>
%H A000225 Wikipedia, <a href="http://en2.wikipedia.org/wiki/Tower_of_Hanoi">Tower
of Hanoi</a>
%H A000225 K. K. Wong, <a href="http://www.lhs.berkeley.edu/Java/Tower/Tower.html">
Tower Of Hanoi:Online Game</a>
%H A000225 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000225 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%F A000225 G.f.: x/((1-2*x)*(1-x)). E.g.f. if offset 1: ((exp(x)-1)^2)/2.
%F A000225 a(n)=sum{k=0..n-1, 2^k} - Paul Barry (pbarry(AT)wit.ie), May 26 2003
%F A000225 a(n)=a(n-1)+2a(n-2)+2, a(0)=0, a(1)=1. - Paul Barry (pbarry(AT)wit.ie),
Jun 06 2003
%F A000225 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
%F A000225 a(n+1) = 2*a(n) + 1, a(0) = 0.
%F A000225 Sum_{k=1..n} C(n, k).
%F A000225 a(n) = n + sum(i=0, n-1, a(i)); a(0) = 0. - Rick L. Shepherd (rshepherd2(AT)hotmail.com),
Aug 04 2004
%F A000225 a(n+1)=(n+1)sum{k=0..n, binomial(n, k)/(k+1)} - Paul Barry (pbarry(AT)wit.ie),
Aug 06 2004
%F A000225 a(n+1)=sum{k=0..n, binomial(n+1, k+1)} - Paul Barry (pbarry(AT)wit.ie),
Aug 23 2004
%F A000225 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
%F A000225 a(n) = A119258(n,n-1) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
May 11 2006
%F A000225 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
%F A000225 Sum_{n=1..inf}1/a(n) = 1,606695152...(Erdos-Borwein constant;see A065442,
A038631) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 27 2006
%F A000225 Stirling_2[n-k,2] starting from n=k+1. - Artur Jasinski (grafix(AT)csl.pl),
Nov 18 2006
%F A000225 a(n) = A125118(n,1) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Nov 21 2006
%F A000225 a(n) = StirlingS2(n+1,2) - Ross La Haye (rlahaye(AT)new.rr.com), Jan
10 2008
%F A000225 a(n) = A024088(n)/A001576(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Feb 15 2009]
%p A000225 A000225 := n->2^n-1; [ seq(2^n-1,n=0..50) ];
%p A000225 seq(add(binomial(n, k)*(bell(k-n)), k=1..n), n=0..32); - Zerinvary Lajos
(zerinvarylajos(AT)yahoo.com), Dec 01 2006
%p A000225 [seq (stirling2(n,2),n=1..33)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo),
Dec 06 2006
%p A000225 a:=n->sum (2^j,j=0..n): seq(a(n),n=-1..31); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jun 27 2007
%p A000225 A000225:=1/(2*z-1)/(z-1); [S. Plouffe in his 1992 dissertation, sequence
starting at a(1).]
%p A000225 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
%p A000225 with(finance):seq(add(futurevalue( 1, 1, k),k=0..n),n=- 1..31); - Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Jun 26 2008
%p A000225 a:=n->(sum((stirling2(n,2)), j=1..n)):seq(a(n), n=0..40): b:=n->(sum((stirling2(n,
2)), j=0..n)):seq(b(n), n=0..40): c:=b-a:seq(c(n), n=1..33); [From
Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 24 2008]
%t A000225 a[n_] := 2^n - 1; Table[a[n], {n, 0, 30}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com),
Mar 30 2006
%t A000225 Array[2^# - 1 &, 50, 0] - Joseph Biberstine (jrbibers(AT)indiana.edu),
Dec 26 2006
%t A000225 a=0;lst={a};Do[a=a*2+1;AppendTo[lst,a],{n,5!}];lst [From Vladimir Orlovsky
(4vladimir(AT)gmail.com), Jan 03 2009]
%t A000225 Table[Sum[ Binomial[n + 1, k + 1], {k, 0, n}], {n, -1, 31}] [From Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Jul 08 2009]
%o A000225 sage: [stirling_number2(i,2) for i in xrange(1,30)] - Zerinvary Lajos
(zerinvarylajos(AT)yahoo.com), Jun 26 2008
%o A000225 (Other) sage: [lucas_number1(n,3,2) for n in xrange(0, 33)]# [From Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]
%o A000225 (Other) sage: [gaussian_binomial(n,1,2) for n in xrange(1,33)] # [From
Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 24 2009]
%o A000225 (PARI) A000225(n) = 2^n-1 [From Michael Porter (michael_b_porter(AT)yahoo.com),
Oct 27 2009]
%Y A000225 Cf. A000079, A016189.
%Y A000225 Cf. a(n)=A112492(n, 2). Rightmost column of A008969.
%Y A000225 a(n) = A118654(n, 1) = A118654(n-1, 3), for n > 0.
%Y A000225 Subsequence of A132781.
%Y A000225 Cf. A000043, A000668. [From Omar E. Pol (info(AT)polprimos.com), Aug
31 2008]
%Y A000225 Cf. A000040, A001348. [From Omar E. Pol (info(AT)polprimos.com), Sep
05 2008]
%Y A000225 A009545, A144081, [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 10
2008]
%Y A000225 A001045 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 23 2009]
%Y A000225 Sequence in context: A097002 A060152 A126646 this_sequence A123121 A117060
A057613
%Y A000225 Adjacent sequences: A000222 A000223 A000224 this_sequence A000226 A000227
A000228
%K A000225 nonn,easy,core,nice,new
%O A000225 0,3
%A A000225 N. J. A. Sloane (njas(AT)research.att.com).
%E A000225 Additional links provided by Lekraj Beedassy (blekraj(AT)yahoo.com),
Dec 20 2003
%E A000225 Removed conjectural attribute from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl),
Mar 11 2009
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