Search: id:A000225 Results 1-1 of 1 results found. %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, Table of n, a(n) for n = 0..1000 %H A000225 Anonymous, The Tower of Hanoi %H A000225 J. Bernheiden, Mersenne Numbers (Text in German) %H A000225 R. P. Brent and H. J. J. te Riele, Factorizations of a^n +- 1, 13 =< a < 100 %H A000225 R. P. Brent, P. L. Montgomery and H. J. J. te Riele, Factorizations of a^n +- 1, 13 =< a < 100 :Update 2 %H A000225 R. P. Brent, P. L. Montgomery and H. J. J. te Riele, Factorizations Of Cunningham Numbers With Bases 13 To 99. Millennium Edition %H A000225 R. P. Brent, P. L. Montgomery and H. J. J. te Riele, Factorizations of Cunningham numbers with bases 13 to 99: Millennium edition %H A000225 R. P. Brent and H. J. J. te Riele, Factorizations of a^n +- 1, 13 =< a <100 %H A000225 John Brillhart et al., Cunningham Project [Factorizations of b^n +- 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers] %H A000225 J. Britton, The Tower of Hanoi %H A000225 C. K. Caldwell, The Prime Glossary, Mersenne number %H A000225 P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. %H A000225 W. M. B. Dukes, On the number of matroids on a finite set %H A000225 W. Edgington, Mersenne Page %H A000225 T. Eveilleau, Animated solution to the Tower of Hanoi problem %H A000225 G. Everest, S. Stevens, D. Tamsett and T. Ward, Primitive divisors of quadratic polynomial sequences %H A000225 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 %H A000225 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 138 %H A000225 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 345 %H A000225 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 371 %H A000225 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 880 %H A000225 J. Loy, The Tower of Hanoi %H A000225 Mathforum, Tower of Hanoi %H A000225 Mathforum, Problem of the Week, The Tower of Hanoi Puzzle %H A000225 NationMaster.com, Tower of Hanoi %H A000225 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. %H A000225 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A000225 Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1. %H A000225 R. R. Snapp, The Tower of Hanoi %H A000225 Thesaurus.maths.org, Mersenne Number %H A000225 Thinks.com, Tower of Hanoi, A classic puzzle game %H A000225 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A000225 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A000225 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A000225 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A000225 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics %H A000225 Eric Weisstein's World of Mathematics, Run %H A000225 Eric Weisstein's World of Mathematics, Rule 222 %H A000225 Wikipedia, Tower of Hanoi %H A000225 K. K. Wong, Tower Of Hanoi:Online Game %H A000225 Index entries for "core" sequences %H A000225 Index entries for sequences related to linear recurrences with constant coefficients %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 Search completed in 0.005 seconds