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A051049 - OEIS
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <html> <head> <link rel="stylesheet" href="/styles.css"> <meta name="format-detection" content="telephone=no"> <meta http-equiv="content-type" content="text/html; charset=utf-8"> <meta name=viewport content="width=device-width, initial-scale=1"> <meta name="keywords" content="OEIS,integer sequences,Sloane" /> <title>A051049 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A051049" function redir() { var host = document.location.hostname; if(host != "oeis.org" && host != "127.0.0.1" && !/^([0-9.]+)$/.test(host) && host != "localhost" && host != "localhost.localdomain") { document.location = "https"+":"+"//"+"oeis"+".org/" + myURL; } } function sf() { if(document.location.pathname == "/" && document.f) document.f.q.focus(); } </script> </head> <body bgcolor=#ffffff onload="redir();sf()"> <div class=loginbar> <div class=login> <a href="/login?redirect=%2fA051049">login</a> </div> </div> <div class=center><div class=top> <center> <div class=donors> The OEIS is supported by <a href="http://oeisf.org/#DONATE">the many generous donors to the OEIS Foundation</a>. </div> <div class=banner> <a href="/"><img class=banner border="0" width="600" src="/banner2021.jpg" alt="A051049 - OEIS"></a> </div> </center> </div></div> <div class=center><div class=pagebody> <div class=searchbarcenter> <form name=f action="/search" method="GET"> <div class=searchbargreet> <div class=searchbar> <div class=searchq> <input class=searchbox maxLength=1024 name=q value="" title="Search Query"> </div> <div class=searchsubmit> <input type=submit value="Search" name=go> </div> <div class=hints> <span class=hints><a href="/hints.html">Hints</a></span> </div> </div> <div class=searchgreet> (Greetings from <a href="/welcome">The On-Line Encyclopedia of Integer Sequences</a>!) </div> </div> </form> </div> <div class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> A051049 </div> <div class=seqname> Number of moves needed to solve an (n+1)-ring baguenaudier if two simultaneous moves of the two end rings are counted as one. </div> </div> <div class=scorerefs> 20 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 1, 4, 7, 16, 31, 64, 127, 256, 511, 1024, 2047, 4096, 8191, 16384, 32767, 65536, 131071, 262144, 524287, 1048576, 2097151, 4194304, 8388607, 16777216, 33554431, 67108864, 134217727, 268435456, 536870911, 1073741824</div> <div class=seqdatalinks> (<a href="/A051049/list">list</a>; <a href="/A051049/graph">graph</a>; <a href="/search?q=A051049+-id:A051049">refs</a>; <a href="/A051049/listen">listen</a>; <a href="/history?seq=A051049">history</a>; <a href="/search?q=id:A051049&fmt=text">text</a>; <a href="/A051049/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,3</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>Might be called the "Purkiss sequence", after Henry John Purkiss who in 1865 found that this is the number of moves for the accelerated Chinese Rings puzzle (baguenaudier). [Email from <a href="/wiki/User:Andreas_M._Hinz">Andreas M. Hinz</a>, Feb 15 2017, who also pointed out that there was an error in the definition in this entry]. - <a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a>, Feb 18 2017</div> <div class=sectline>The row sums of triangle <a href="/A166692" title="Triangle T(n,k) read by rows: T(n,k) = 2^(k-1), k>0, T(n,0) = (n+1) mod 2.">A166692</a>. - <a href="/wiki/User:Paul_Curtz">Paul Curtz</a>, Oct 20 2009</div> <div class=sectline>The inverse binomial transform equals (-1)^n*<a href="/A062510" title="a(n) = 2^n + (-1)^(n+1).">A062510</a>(n) with an extra leading term 1. - <a href="/wiki/User:Paul_Curtz">Paul Curtz</a>, Oct 20 2009</div> <div class=sectline>This is the sequence A(1,1;1,2;1) 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. - <a href="/wiki/User:Wolfdieter_Lang">Wolfdieter Lang</a>, Oct 18 2010</div> <div class=sectline>Also, the decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by Rules 261, 269, 277, 285, 293, 301, 309, 317, 325, 333, 341, 349, 357, 365, 37, and 381, based on the 5-celled von Neumann neighborhood. - <a href="/wiki/User:Robert_Price">Robert Price</a>, Jan 02 2017</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Vincenzo Librandi, <a href="/A051049/b051049.txt">Table of n, a(n) for n = 0..1000</a></div> <div class=sectline>Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19, 2016, #16.3.5.</div> <div class=sectline>Andreas M. Hinz, <a href="https://www.fq.math.ca/Papers1/55-1/Hinz09152016.pdf">The Lichtenberg sequence</a>, Fib. Quart., 55 (2017), 2-12.</div> <div class=sectline>A. M. Hinz, S. Klav啪ar, U. Milutinovi膰, and C. Petr, <a href="http://dx.doi.org/10.1007/978-3-0348-0237-6">The Tower of Hanoi - Myths and Maths</a>, Birkh盲user 2013. See page 56. <a href="http://tohbook.info">Book's website</a></div> <div class=sectline>Wolfdieter Lang, <a href="/A051049/a051049.pdf">Notes on certain inhomogeneous three term recurrences.</a> [From <a href="/wiki/User:Wolfdieter_Lang">Wolfdieter Lang</a>, Oct 18 2010]</div> <div class=sectline>N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015.</div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Baguenaudier.html">Baguenaudier</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a></div> <div class=sectline>S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a></div> <div class=sectline>Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a></div> <div class=sectline><a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a></div> <div class=sectline><a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a></div> <div class=sectline><a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a></div> <div class=sectline><a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2).</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = (2^(n+1) - (1 + (-1)^(n+1)))/2. - <a href="/wiki/User:Paul_Barry">Paul Barry</a>, Apr 24 2003</div> <div class=sectline>a(n+2) = a(n+1) + 2*a(n) + 1, a(0)=a(1)=1. - <a href="/wiki/User:Paul_Barry">Paul Barry</a>, May 01 2003</div> <div class=sectline>From <a href="/wiki/User:Paul_Barry">Paul Barry</a>, Sep 19 2003: (Start)</div> <div class=sectline>G.f.: (1 - x + x^2)/((1 - x^2)*(1 - 2*x));</div> <div class=sectline>e.g.f.: exp(2*x) - sinh(x). (End)</div> <div class=sectline>a(n) = ((Sum_{k=0..n} 2^k) + (-1)^n)/2 = (<a href="/A000225" title="a(n) = 2^n - 1. (Sometimes called Mersenne numbers, although that name is usually reserved for A001348.)">A000225</a>(n+1) + (-1)^n)/2. - <a href="/wiki/User:Paul_Barry">Paul Barry</a>, May 27 2003</div> <div class=sectline>(a(n+1) - a(n))/3 = <a href="/A001045" title="Jacobsthal sequence (or Jacobsthal numbers): a(n) = a(n-1) + 2*a(n-2), with a(0) = 0, a(1) = 1; also a(n) = nearest integer ...">A001045</a>(n). - <a href="/wiki/User:Paul_Barry">Paul Barry</a>, May 27 2003</div> <div class=sectline>a(n) = Sum_{k=0..floor(n/2)} binomial(n+1, 2*k). - <a href="/wiki/User:Paul_Barry">Paul Barry</a>, May 27 2003</div> <div class=sectline>a(n) = (Sum_{k=0..n} binomial(n,k) + (-1)^(n-k)) - 1. - <a href="/wiki/User:Paul_Barry">Paul Barry</a>, Jul 21 2003</div> <div class=sectline>a(n) = Sum_{k=0..n} Sum_{j=0..n-k, (j-k) mod 2 = 0} binomial(n-k, j). - <a href="/wiki/User:Paul_Barry">Paul Barry</a>, Jan 25 2005</div> <div class=sectline>Row sums of triangle <a href="/A135221" title="Triangle A007318 + A000012(signed) - I, I = Identity matrix, read by rows.">A135221</a>. - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Nov 23 2007</div> <div class=sectline>a(n) = <a href="/A001045" title="Jacobsthal sequence (or Jacobsthal numbers): a(n) = a(n-1) + 2*a(n-2), with a(0) = 0, a(1) = 1; also a(n) = nearest integer ...">A001045</a>(n+1) + <a href="/A000975" title="a(2n) = 2*a(2n-1), a(2n+1) = 2*a(2n)+1 (also a(n) is the n-th number without consecutive equal binary digits).">A000975</a>(n+1) - <a href="/A000079" title="Powers of 2: a(n) = 2^n.">A000079</a>(n). - <a href="/wiki/User:Paul_Curtz">Paul Curtz</a>, Oct 20 2009</div> <div class=sectline>a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), a(0) = a(1) = 1, a(2) = 4. Observed by G. Detlefs. See the W. Lang link. - <a href="/wiki/User:Wolfdieter_Lang">Wolfdieter Lang</a>, Oct 18 2010</div> <div class=sectline>a(n) = 3*a(n-1) - 2*a(n-2) + 3*(-1)^n. - <a href="/wiki/User:Gary_Detlefs">Gary Detlefs</a>, Dec 21 2010</div> <div class=sectline>a(n) = 3* <a href="/A000975" title="a(2n) = 2*a(2n-1), a(2n+1) = 2*a(2n)+1 (also a(n) is the n-th number without consecutive equal binary digits).">A000975</a>(n-1) + 1, n > 0. - <a href="/wiki/User:Gary_Detlefs">Gary Detlefs</a>, Dec 21 2010</div> <div class=sectline>a(n+2) = <a href="/A001969" title="Evil numbers: nonnegative integers with an even number of 1's in their binary expansion.">A001969</a>(2^n+1) + <a href="/A000069" title="Odious numbers: numbers with an odd number of 1's in their binary expansion.">A000069</a>(2^n); evil + odious. - <a href="/wiki/User:Johannes_W._Meijer">Johannes W. Meijer</a>, Jun 24 2011, Jun 26 2011</div> <div class=sectline>E.g.f.: exp(2x) - sinh(x) = Q(0); Q(k) = 1 - k!*x^(k+1)/((2*k + 1)!*2^k - 2*(((2*k + 1)!*2^k)^2)/((2*k + 1)!*2^(k+1) - x^k*(k + 1)!/Q(k+1))); (continued fraction). - <a href="/wiki/User:Sergei_N._Gladkovskii">Sergei N. Gladkovskii</a>, Nov 16 2011</div> <div class=sectline>a(n) = Sum_{k=0..n} Sum_{i=0..n} C(k-1,i). - <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Sep 21 2017</div> <div class=sectline>a(n) = <a href="/A000975" title="a(2n) = 2*a(2n-1), a(2n+1) = 2*a(2n)+1 (also a(n) is the n-th number without consecutive equal binary digits).">A000975</a>(n+1) - <a href="/A001045" title="Jacobsthal sequence (or Jacobsthal numbers): a(n) = a(n-1) + 2*a(n-2), with a(0) = 0, a(1) = 1; also a(n) = nearest integer ...">A001045</a>(n). - <a href="/wiki/User:Yuchun_Ji">Yuchun Ji</a>, Jul 08 2018</div> <div class=sectline>a(n) = <a href="/A026147" title="a(n) = position of n-th 1 in A001285 or A010059 (Thue-Morse sequence).">A026147</a>(2^(n-1)) for n > 0. - <a href="/wiki/User:Chunqing_Liu">Chunqing Liu</a>, Dec 18 2022</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline><a href="/A051049" title="Number of moves needed to solve an (n+1)-ring baguenaudier if two simultaneous moves of the two end rings are counted as one.">A051049</a>:= proc(n): 2^n -(1-(-1)^n)/2 end: seq(<a href="/A051049" title="Number of moves needed to solve an (n+1)-ring baguenaudier if two simultaneous moves of the two end rings are counted as one.">A051049</a>(n), n=0..40); # <a href="/wiki/User:Johannes_W._Meijer">Johannes W. Meijer</a>, Jun 24 2011</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>a[n_?EvenQ]:= 2^(n-1) -1; a[n_?OddQ]:= 2^(n-1); Table[a[n], {n, 50}]</div> <div class=sectline>LinearRecurrence[{2, 1, -2}, {1, 1, 4}, 40] (* <a href="/wiki/User:Jean-Fran莽ois_Alcover">Jean-Fran莽ois Alcover</a>, Jan 08 2019 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Magma) [2^n -(1-(-1)^n)/2: n in [0..40]]; // <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Aug 14 2011</div> <div class=sectline>(PARI) a(n)=2^(n-1)-(n%2==0) \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Mar 22 2013</div> <div class=sectline>(SageMath) [2^n -(n%2) for n in range(41)] # <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Apr 23 2023</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A000069" title="Odious numbers: numbers with an odd number of 1's in their binary expansion.">A000069</a>, <a href="/A000079" title="Powers of 2: a(n) = 2^n.">A000079</a>, <a href="/A000225" title="a(n) = 2^n - 1. (Sometimes called Mersenne numbers, although that name is usually reserved for A001348.)">A000225</a>, <a href="/A000975" title="a(2n) = 2*a(2n-1), a(2n+1) = 2*a(2n)+1 (also a(n) is the n-th number without consecutive equal binary digits).">A000975</a>, <a href="/A001045" title="Jacobsthal sequence (or Jacobsthal numbers): a(n) = a(n-1) + 2*a(n-2), with a(0) = 0, a(1) = 1; also a(n) = nearest integer ...">A001045</a>.</div> <div class=sectline>Cf. <a href="/A001969" title="Evil numbers: nonnegative integers with an even number of 1's in their binary expansion.">A001969</a>, <a href="/A026147" title="a(n) = position of n-th 1 in A001285 or A010059 (Thue-Morse sequence).">A026147</a>, <a href="/A062510" title="a(n) = 2^n + (-1)^(n+1).">A062510</a>, <a href="/A135221" title="Triangle A007318 + A000012(signed) - I, I = Identity matrix, read by rows.">A135221</a>.</div> <div class=sectline>Row sums of <a href="/A131086" title="Triangle read by rows: T(n,k) = 2*binomial(n,k) - (-1)^(n-k) (0 <= k <= n).">A131086</a>.</div> <div class=sectline>Row sums of <a href="/A166692" title="Triangle T(n,k) read by rows: T(n,k) = 2^(k-1), k>0, T(n,0) = (n+1) mod 2.">A166692</a>.</div> <div class=sectline>Sequence in context: <a href="/A286741" title="Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cell...">A286741</a> <a href="/A298344" title="a(n) = a(n-1) + a(n-2) + a([n/3]) + a([2n/3]), where a(0) = 1, a(1) = 1, a(2) = 1.">A298344</a> <a href="/A285654" title="Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cell...">A285654</a> * <a href="/A298415" title="a(n) = a(n-1) + 2*a(n-2) + a([n/2]), where a(0) = 1, a(1) = 1, a(2) = 1.">A298415</a> <a href="/A373653" title="a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-5*k-1,k).">A373653</a> <a href="/A108122" title="G.f.: (1-2*x^2)/(1-x-2*x^2-x^3).">A108122</a></div> <div class=sectline>Adjacent sequences: <a href="/A051046" title="Numbers k for which pi(k) > k/(H_k - 3/2), where pi is the prime-counting function and H_k is the k-th harmonic number.">A051046</a> <a href="/A051047" title="For n > 5, a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3); initial terms are 1, 3, 8, 120, 1680.">A051047</a> <a href="/A051048" title="Sqrt[a(n)a(n+1)+1] of A051047.">A051048</a> * <a href="/A051050" title="Numerator of the probability that the convex hull of n+2 randomly chosen points in the unit ball B^n has n+1 vertices.">A051050</a> <a href="/A051051" title="Denominators of the probability that the convex hull of n+2 randomly chosen points in the unit ball B^n has n+1 vertices (wi...">A051051</a> <a href="/A051052" title="a(n) = binomial(n, floor(n/5)).">A051052</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="a sequence of nonnegative numbers">nonn</span>,<span title="it is very easy to produce terms of sequence">easy</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Eric_W._Weisstein">Eric W. Weisstein</a></div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>Edited and information added by <a href="/wiki/User:Johannes_W._Meijer">Johannes W. Meijer</a>, Jun 24 2011</div> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <div class=sectline>approved</div> </div> </div> </div> <div class=space10></div> </div> </div></div> <p> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome"><font color="red">Welcome</font></a> <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> <a href="/wiki/Special:RequestAccount">Register</a> <a href="/play.html">Music</a> <a href="/plot2.html">Plot 2</a> <a href="/demo1.html">Demos</a> <a href="/wiki/Index_to_OEIS">Index</a> <a href="/webcam">WebCam</a> <a href="/Submit.html">Contribute</a> <a href="/eishelp2.html">Format</a> <a href="/wiki/Style_Sheet">Style Sheet</a> <a href="/transforms.html">Transforms</a> <a href="/ol.html">Superseeker</a> <a href="/recent">Recents</a> </div> <div class=linksbar> <a href="/community.html">The OEIS Community</a> </div> <div class=linksbar> Maintained by <a href="http://oeisf.org">The OEIS Foundation Inc.</a> </div> <div class=dbinfo>Last modified December 1 04:55 EST 2024. 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