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A001951 - 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>A001951 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A001951" 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=%2fA001951">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="A001951 - 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> A001951 </div> <div class=seqname> A Beatty sequence: a(n) = floor(n*sqrt(2)). <br><font size=-1>(Formerly M0955 N0356)</font> </div> </div> <div class=scorerefs> 141 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>0, 1, 2, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, 22, 24, 25, 26, 28, 29, 31, 32, 33, 35, 36, 38, 39, 41, 42, 43, 45, 46, 48, 49, 50, 52, 53, 55, 56, 57, 59, 60, 62, 63, 65, 66, 67, 69, 70, 72, 73, 74, 76, 77, 79, 80, 82, 83, 84, 86, 87, 89, 90, 91, 93, 94, 96, 97, 98, 100</div> <div class=seqdatalinks> (<a href="/A001951/list">list</a>; <a href="/A001951/graph">graph</a>; <a href="/search?q=A001951+-id:A001951">refs</a>; <a href="/A001951/listen">listen</a>; <a href="/history?seq=A001951">history</a>; <a href="/search?q=id:A001951&fmt=text">text</a>; <a href="/A001951/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>Earliest monotonic sequence greater than 0 satisfying the condition: "a(n) + 2n is not in the sequence". - <a href="/wiki/User:Benoit_Cloitre">Benoit Cloitre</a>, Mar 25 2004</div> <div class=sectline>Also the integer part of the hypotenuse of isosceles right triangles. The real part of these numbers is irrational. For proof see Jones and Jones.</div> <div class=sectline>First differences are 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, ... (<a href="/A006337" title="An "eta-sequence": a(n) = floor( (n+1)*sqrt(2) ) - floor( n*sqrt(2) ).">A006337</a> with a 1 in front). - <a href="/wiki/User:Philippe_Del茅ham">Philippe Del茅ham</a>, May 29 2006</div> <div class=sectline>It appears that the distance between the a(n)-th triangular number and the nearest square is not greater than floor(a(n)/2). - <a href="/wiki/User:Ralf_Stephan">Ralf Stephan</a>, Sep 14 2013</div> <div class=sectline>These are the nonnegative integers m satisfying sin(m*Pi/r)*sin((m+1)*Pi/r) <= 0, where r = sqrt(2). In general, the Beatty sequence of an irrational number r > 1 consists of the numbers m satisfying sin(m*x)*sin((m+1)*x) <= 0, where x = Pi/r. Thus the numbers m satisfying sin(m*x)*sin((m+1)*x) > 0 form the Beatty sequence of r/(1-r). - <a href="/wiki/User:Clark_Kimberling">Clark Kimberling</a>, Aug 21 2014</div> <div class=sectline>For n > 0: <a href="/A080764" title="First differences of A049472, floor(n/sqrt(2)).">A080764</a>(a(n)) = 1. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jul 03 2015</div> <div class=sectline>From <a href="/wiki/User:Clark_Kimberling">Clark Kimberling</a>, Oct 17 2016: (Start)</div> <div class=sectline>We can generate <a href="/A001951" title="A Beatty sequence: a(n) = floor(n*sqrt(2)).">A001951</a> and <a href="/A001952" title="A Beatty sequence: a(n) = floor(n*(2 + sqrt(2))).">A001952</a> without using sqrt(2).</div> <div class=sectline>First write the even positive integers in a row:</div> <div class=sectline> 2 4 6 8 10 12 14 . . .</div> <div class=sectline>Then put 1 under 2 and add:</div> <div class=sectline> 2 4 6 8 10 12 14 . . .</div> <div class=sectline> 1</div> <div class=sectline> 3</div> <div class=sectline>Next, under 4, put the least positive integer that is not yet in rows 2 and 3;</div> <div class=sectline>it is 2; and add:</div> <div class=sectline> 2 4 6 8 10 12 14 . . .</div> <div class=sectline> 1 2</div> <div class=sectline> 3 6</div> <div class=sectline>Next, under the 6 in row 1, put the least positive integer not yet in rows 2 and 3;</div> <div class=sectline>it is 4, and add:</div> <div class=sectline> 2 4 6 8 10 12 14 . . .</div> <div class=sectline> 1 2 4</div> <div class=sectline> 3 6 10</div> <div class=sectline>Continue in this manner. (End)</div> <div class=sectline>This sequence contains an infinite number of powers of 2 (proof in Crux Mathematicorum link). See <a href="/A103341" title="Numbers k such that floor(k*sqrt(2)) is a power of 2.">A103341</a>. - <a href="/wiki/User:Bernard_Schott">Bernard Schott</a>, Mar 08 2019</div> <div class=sectline>The terms of this sequence generate the multiplicative group of positive rational numbers (observation by Stephen M. Gagola, Jr.; see References). - <a href="/wiki/User:Allen_Stenger">Allen Stenger</a>, Aug 05 2023</div> <div class=sectline>a(n) is also the number of distinct straight cylinders with integer radius and height having the same surface as a sphere with radius n. - <a href="/wiki/User:Felix_Huber">Felix Huber</a>, Sep 20 2024</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>Eric Duch锚ne, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, Urban Larsson, Wythoff Visions, Games of No Chance, Vol. 5; MSRI Publications, Vol. 70 (2017), pages 101-153.</div> <div class=sectline>Stephen M. Gagola Jr., Solution of Problem 12282, Am. Math. Monthly, 130 (2023), pp. 682-683.</div> <div class=sectline>R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 77.</div> <div class=sectline>Gareth A. Jones and J. Mary Jones, Elementary Number Theory, Springer, 1998; pp. 221-222.</div> <div class=sectline>N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).</div> <div class=sectline>N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).</div> <div class=sectline>Roland Sprague, Recreations in Mathematics, Blackie and Son, (1963).</div> <div class=sectline>David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition (1997), Entry sqrt(2), p. 18.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Vincenzo Librandi, <a href="/A001951/b001951.txt">Table of n, a(n) for n = 0..10000</a></div> <div class=sectline>L. Carlitz, Richard Scoville and Verner E. Hoggatt, Jr., <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/10-5/carlitz1.pdf">Pellian representatives</a>, Fib. Quart., 10 (1972), 449-488.</div> <div class=sectline>Ed Doolittle, <a href="https://cms.math.ca/crux/backfile/Crux_v14n03_Mar.pdf">Problem 19</a>, 26th I.M.O. Finland proposed by Romania, Crux Mathematicorum, p. 70, Vol. 14, Mar. 88.</div> <div class=sectline>Ian G. Connell, <a href="https://dx.doi.org/10.4153/CMB-1959-024-3">A generalization of Wythoff's game</a>, Canad. Math. Bull. 2 (1959) 181-190</div> <div class=sectline>Aviezri S. Fraenkel, <a href="http://www.jstor.org/stable/2321643">How to beat your Wythoff games' opponent on three fronts</a>, Amer. Math. Monthly, 89 (1982), 353-361 (the case a=2).</div> <div class=sectline>Aviezri S. Fraenkel, <a href="https://dx.doi.org/10.1016/S0012-365X(00)00138-2">On the recurrence f(m+1)= b(m)*f(m)-f(m-1) and applications</a>, Discrete Mathematics 224 (2000), no. 1-3, pp. 273-279.</div> <div class=sectline>Wen An Liu and Xiao Zhao, <a href="https://dx.doi.org/10.1016/j.dam.2014.08.009">Adjoining to (s,t)-Wythoff's game its P-positions as moves</a>, Discrete Applied Mathematics, Aug 27 2014; See Table 3.</div> <div class=sectline>Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, <a href="https://arxiv.org/abs/2402.08331">Beatty Sequences for a Quadratic Irrational: Decidability and Applications</a>, arXiv:2402.08331 [math.NT], 2024. See pp. 17-18.</div> <div class=sectline>N. J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence)</div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BeattySequence.html">Beatty Sequence.</a></div> <div class=sectline><a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = <a href="/A000196" title="Integer part of square root of n. Or, number of positive squares <= n. Or, n appears 2n+1 times.">A000196</a>(<a href="/A001105" title="a(n) = 2*n^2.">A001105</a>(n)). - <a href="/wiki/User:Jason_Kimberley">Jason Kimberley</a>, Oct 26 2016</div> <div class=sectline>a(n) = floor(csc(1/(sqrt(2)*n))) for n > 0, since sqrt(2)*n < csc(1/(sqrt(2)*n)) < sqrt(2)*n + 1/(3*sqrt(2)*n) < floor(sqrt(2)*n) + 1 for n > 0. - <a href="/wiki/User:Jianing_Song">Jianing Song</a>, Sep 07 2021</div> <div class=sectline>a(n) = <a href="/A194102" title="a(n) = Sum_{j=1..n} floor(j*sqrt(2)); n-th partial sum of Beatty sequence for sqrt(2), A001951.">A194102</a>(n) - <a href="/A194102" title="a(n) = Sum_{j=1..n} floor(j*sqrt(2)); n-th partial sum of Beatty sequence for sqrt(2), A001951.">A194102</a>(n-1) for n > 0. - <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Apr 23 2022</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline>a:=n->floor(n*sqrt(2)): seq(a(n), n=0..80); # <a href="/wiki/User:Muniru_A_Asiru">Muniru A Asiru</a>, Mar 09 2019</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Floor[Range[0, 72] Sqrt[2]] (* <a href="/wiki/User:Robert_G._Wilson_v">Robert G. Wilson v</a>, Oct 17 2012 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) f(n) = for(j=1, n, print1(floor(sqrt(2*j^2))", "))</div> <div class=sectline>(PARI) a(n)=sqrtint(2*n^2) \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Oct 19 2016</div> <div class=sectline>(Magma) [Floor(n*Sqrt(2)): n in [0..60]]; // <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Oct 22 2011</div> <div class=sectline>(Magma) [Isqrt(2*n^2):n in[0..60]]; // <a href="/wiki/User:Jason_Kimberley">Jason Kimberley</a>, Oct 28 2016</div> <div class=sectline>(Maxima) makelist(floor(n*sqrt(2)), n, 0, 100); /* <a href="/wiki/User:Martin_Ettl">Martin Ettl</a>, Oct 17 2012 */</div> <div class=sectline>(Haskell)</div> <div class=sectline>a001951 = floor . (* sqrt 2) . fromIntegral</div> <div class=sectline>-- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Sep 14 2014</div> <div class=sectline>(Python)</div> <div class=sectline>from sympy import integer_nthroot</div> <div class=sectline>def <a href="/A001951" title="A Beatty sequence: a(n) = floor(n*sqrt(2)).">A001951</a>(n): return integer_nthroot(2*n**2, 2)[0] # <a href="/wiki/User:Chai_Wah_Wu">Chai Wah Wu</a>, Mar 16 2021</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Complement of <a href="/A001952" title="A Beatty sequence: a(n) = floor(n*(2 + sqrt(2))).">A001952</a>. Equals <a href="/A001952" title="A Beatty sequence: a(n) = floor(n*(2 + sqrt(2))).">A001952</a>(n) - 2*n for n>0.</div> <div class=sectline>Equals <a href="/A003151" title="Beatty sequence for 1+sqrt(2); a(n) = floor(n*(1+sqrt(2))).">A003151</a>(n) - n; a bisection of <a href="/A094077" title="a(1)=1 and, for n>1, a(n)=a(n-1)+n if n is even and a(n)=smallest positive integer which has not yet appeared in the sequenc...">A094077</a>.</div> <div class=sectline>Bisections: <a href="/A022842" title="Beatty sequence for sqrt(8).">A022842</a>, <a href="/A342281" title="A bisection of A001951: a(n) = A001951(2*n+1).">A342281</a>.</div> <div class=sectline>Cf. <a href="/A022342" title="Integers with "even" Zeckendorf expansions (do not end with ...+F_2 = ...+1) (the Fibonacci-even numbers); also, apart from ...">A022342</a>, <a href="/A026250" title="Beginning with the natural numbers, swap [ k*sqrt(2) ] and [ k*(2 + sqrt(2)) ], for all k >= 1.">A026250</a>, <a href="/A080764" title="First differences of A049472, floor(n/sqrt(2)).">A080764</a>, <a href="/A103341" title="Numbers k such that floor(k*sqrt(2)) is a power of 2.">A103341</a>.</div> <div class=sectline>The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with <a href="/A003151" title="Beatty sequence for 1+sqrt(2); a(n) = floor(n*(1+sqrt(2))).">A003151</a> as the parent: <a href="/A003151" title="Beatty sequence for 1+sqrt(2); a(n) = floor(n*(1+sqrt(2))).">A003151</a>, <a href="/A001951" title="A Beatty sequence: a(n) = floor(n*sqrt(2)).">A001951</a>, <a href="/A001952" title="A Beatty sequence: a(n) = floor(n*(2 + sqrt(2))).">A001952</a>, <a href="/A003152" title="A Beatty sequence: a(n) = floor(n*(1+1/sqrt(2))).">A003152</a>, <a href="/A006337" title="An "eta-sequence": a(n) = floor( (n+1)*sqrt(2) ) - floor( n*sqrt(2) ).">A006337</a>, <a href="/A080763" title="Exchange 1's and 2's in the eta-sequence A006337.">A080763</a>, <a href="/A082844" title="Start with 3,2 and apply the rule a(a(1)+a(2)+...+a(n)) = a(n), fill in any undefined terms with a(t) = 2 if a(t-1) = 3 and ...">A082844</a> (conjectured), <a href="/A097509" title="a(n) is the number of times that n occurs as floor(k * sqrt(2)) - k.">A097509</a>, <a href="/A159684" title="Sturmian word: limit S(infinity) where S(0) = 0, S(1) = 0,1 and for n>=1, S(n+1) = S(n)S(n)S(n-1).">A159684</a>, <a href="/A188037" title="a(n) = floor(nr) - 1 - floor((n-1)r), where r = sqrt(2).">A188037</a>, <a href="/A245219" title="Continued fraction expansion of the constant c in A245218; c = sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x +...">A245219</a> (conjectured), <a href="/A276862" title="First differences of the Beatty sequence A003151 for 1 + sqrt(2).">A276862</a>. - <a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a>, Mar 09 2021</div> <div class=sectline>Partial sums: <a href="/A194102" title="a(n) = Sum_{j=1..n} floor(j*sqrt(2)); n-th partial sum of Beatty sequence for sqrt(2), A001951.">A194102</a>.</div> <div class=sectline>Sequence in context: <a href="/A258833" title="Nonhomogeneous Beatty sequence: ceiling((n + 1/4)*sqrt(2)).">A258833</a> <a href="/A097506" title="Duplicate of A001951.">A097506</a> <a href="/A189794" title="(A189792)/2.">A189794</a> * <a href="/A039046" title="Numbers whose base-7 representation has the same number of 3's and 6's.">A039046</a> <a href="/A187683" title="Rank transform of the sequence floor(2n/3); complement of A187683.">A187683</a> <a href="/A187351" title="Rank transform of the sequence floor(n/sqrt(2)); complement of A187352.">A187351</a></div> <div class=sectline>Adjacent sequences: <a href="/A001948" title="These numbers when multiplied by all powers of 4 give the numbers that are not the sums of 4 distinct squares.">A001948</a> <a href="/A001949" title="Solutions of a fifth-order probability difference equation.">A001949</a> <a href="/A001950" title="Upper Wythoff sequence (a Beatty sequence): a(n) = floor(n*phi^2), where phi = (1+sqrt(5))/2.">A001950</a> * <a href="/A001952" title="A Beatty sequence: a(n) = floor(n*(2 + sqrt(2))).">A001952</a> <a href="/A001953" title="a(n) = floor((n + 1/2) * sqrt(2)).">A001953</a> <a href="/A001954" title="a(n) = floor((n+1/2)*(2+sqrt(2))); winning positions in the 2-Wythoff game.">A001954</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="an exceptionally nice sequence">nice</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:N._J._A._Sloane">N. J. A. Sloane</a></div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>More terms from <a href="/wiki/User:David_W._Wilson">David W. Wilson</a>, Sep 20 2000</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 April 7 00:47 EDT 2025. 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