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A080764 - 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>A080764 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A080764" 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=%2fA080764">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="A080764 - 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> A080764 </div> <div class=seqname> First differences of <a href="/A049472" title="a(n) = floor(n/sqrt(2)).">A049472</a>, floor(n/sqrt(2)). </div> </div> <div class=scorerefs> 40 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0</div> <div class=seqdatalinks> (<a href="/A080764/list">list</a>; <a href="/A080764/graph">graph</a>; <a href="/search?q=A080764+-id:A080764">refs</a>; <a href="/A080764/listen">listen</a>; <a href="/history?seq=A080764">history</a>; <a href="/search?q=id:A080764&fmt=text">text</a>; <a href="/A080764/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,1</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>Fixed point of the morphism 0->1, 1->110. - <a href="/wiki/User:Benoit_Cloitre">Benoit Cloitre</a>, May 31 2004</div> <div class=sectline>As binary constant 0.1101101110110... = 0.85826765646... (<a href="/A119812" title="Decimal expansion of the constant defined by binary sums involving Beatty sequences: c = Sum_{n>=1} A049472(n)/2^n = Sum_{n>...">A119812</a>), see Fxtbook link. - <a href="/wiki/User:Joerg_Arndt">Joerg Arndt</a>, May 15 2011</div> <div class=sectline>Characteristic word with slope 1/sqrt(2) [see J. L. Ramirez et al.]. - <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Jul 09 2013</div> <div class=sectline>From <a href="/wiki/User:Peter_Bala">Peter Bala</a>, Nov 22 2013: (Start)</div> <div class=sectline>Sturmian word: equals the limit word S(infinity) where S(0) = 0, S(1) = 1 and for n >= 1, S(n+1) = S(n)S(n)S(n-1).</div> <div class=sectline>More generally, for k = 0,1,2,..., we can define a sequence of words S_k(n) by S_k(0) = 0, S_k(1) = 0...01 (k 0's) and for n >= 1, S_k(n+1) = S_k(n)S_k(n)S_k(n-1). Then the limit word S_k(infinity) is a Sturmian word whose terms are given by a(n) = floor((n + 2)/(k + sqrt(2))) - floor((n + 1)/(k + sqrt(2))).</div> <div class=sectline>This sequence corresponds to the case k = 0. See <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> (case k = 1) and <a href="/A171588" title="The Pell word: Fixed point of the morphism 0->001, 1->0.">A171588</a> (case k = 2). Compare with the Fibonacci words <a href="/A005614" title="The binary complement of the infinite Fibonacci word A003849. Start with 1, apply 0->1, 1->10, iterate, take limit.">A005614</a>, <a href="/A221150" title="The generalized Fibonacci word f^[3].">A221150</a>, <a href="/A221151" title="The generalized Fibonacci word f^[4].">A221151</a> and <a href="/A221152" title="The generalized Fibonacci word f^[5].">A221152</a>. See also <a href="/A230901" title="Sturmian word: equals the limit word S(infinity) where S(0) = 0, S(1) = 1 and for n >= 1, S(n+1) = S(n)S(n)S(n)S(n-1).">A230901</a>. (End)</div> <div class=sectline>For n > 0: a(<a href="/A001951" title="A Beatty sequence: a(n) = floor(n*sqrt(2)).">A001951</a>(n)) = 1, a(<a href="/A001952" title="A Beatty sequence: a(n) = floor(n*(2 + sqrt(2))).">A001952</a>(n)) = 0. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jul 03 2015</div> <div class=sectline>Binary complement of the Pell word <a href="/A171588" title="The Pell word: Fixed point of the morphism 0->001, 1->0.">A171588</a>. - <a href="/wiki/User:Michel_Dekking">Michel Dekking</a>, Feb 22 2018</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Reinhard Zumkeller, <a href="/A080764/b080764.txt">Table of n, a(n) for n = 0..10000</a></div> <div class=sectline>Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 38.12, pp. 757-758.</div> <div class=sectline>Wikipedia, <a href="https://en.wikipedia.org/wiki/Sturmian_word">Sturmian word</a></div> <div class=sectline><a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = floor((n+2)*sqrt(2)/2) - floor((n+1)*sqrt(2)/2).</div> <div class=sectline>a(n) = <a href="/A188295" title="[nr]-[nr-r], where r=1/sqrt(2), [ ]=floor.">A188295</a>(n+2) for all n in Z. - <a href="/wiki/User:Michael_Somos">Michael Somos</a>, Aug 19 2018</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>From <a href="/wiki/User:Peter_Bala">Peter Bala</a>, Nov 22 2013: (Start)</div> <div class=sectline>The first few Sturmian words S(n) are</div> <div class=sectline>S(0) = 0</div> <div class=sectline>S(1) = 1</div> <div class=sectline>S(2) = 110</div> <div class=sectline>S(3) = 110 110 1</div> <div class=sectline>S(4) = 1101101 1101101 110</div> <div class=sectline>S(5) = 11011011101101110 11011011101101110 1101101</div> <div class=sectline>The lengths of the words are [1, 1, 3, 7, 17, 41, ...] = <a href="/A001333" title="Pell-Lucas numbers: numerators of continued fraction convergents to sqrt(2).">A001333</a>. (End)</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline><a href="/A080764" title="First differences of A049472, floor(n/sqrt(2)).">A080764</a> := proc(n)</div> <div class=sectline> alpha := 1/sqrt(2) ;</div> <div class=sectline> floor((n+2)*alpha)-floor((n+1)*alpha) ;</div> <div class=sectline>end proc: # <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Jul 09 2013</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Nest[ Flatten[ # /. {0 -> 1, 1 -> {1, 1, 0}}] &, {1}, 7] (* <a href="/wiki/User:Robert_G._Wilson_v">Robert G. Wilson v</a>, Apr 16 2005 *)</div> <div class=sectline>NestList[ Flatten[ # /. {0 -> {1}, 1 -> {1, 0, 1}}] &, {1}, 5] // Flatten (* or *)</div> <div class=sectline>t = Table[Floor[n/Sqrt[2]], {n, 111}]; Drop[t, 1] - Drop[t, -1] (* <a href="/wiki/User:Robert_G._Wilson_v">Robert G. Wilson v</a>, Nov 03 2005 *)</div> <div class=sectline>a[ n_] := With[{m = n + 1}, Floor[(m + 1) / Sqrt[2]] - Floor[m / Sqrt[2]]]; (* <a href="/wiki/User:Michael_Somos">Michael Somos</a>, Aug 19 2018 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Haskell)</div> <div class=sectline>a080764 n = a080764_list !! n</div> <div class=sectline>a080764_list = tail $ zipWith (-) (tail a049472_list) a049472_list</div> <div class=sectline>-- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jul 03 2015</div> <div class=sectline>(PARI) {a(n) = n++; my(k = sqrtint(n*n\2)); n*(n+2) > 2*k*(k+2)}; /* <a href="/wiki/User:Michael_Somos">Michael Somos</a>, Aug 19 2018 */</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A005614" title="The binary complement of the infinite Fibonacci word A003849. Start with 1, apply 0->1, 1->10, iterate, take limit.">A005614</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="/A171588" title="The Pell word: Fixed point of the morphism 0->001, 1->0.">A171588</a>, <a href="/A221150" title="The generalized Fibonacci word f^[3].">A221150</a>, <a href="/A221151" title="The generalized Fibonacci word f^[4].">A221151</a>, <a href="/A221152" title="The generalized Fibonacci word f^[5].">A221152</a>, <a href="/A230901" title="Sturmian word: equals the limit word S(infinity) where S(0) = 0, S(1) = 1 and for n >= 1, S(n+1) = S(n)S(n)S(n)S(n-1).">A230901</a>.</div> <div class=sectline>Cf. <a href="/A049472" title="a(n) = floor(n/sqrt(2)).">A049472</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="/A188295" title="[nr]-[nr-r], where r=1/sqrt(2), [ ]=floor.">A188295</a>.</div> <div class=sectline>Sequence in context: <a href="/A015527" title="Inverse of 1518th cyclotomic polynomial.">A015527</a> <a href="/A276395" title="Characteristic function of floor(36*n/25).">A276395</a> <a href="/A232750" title="a(0)=1, after which a(n) = Number of terms of A005228 which occur between each consecutive terms of A232739, in range A23273...">A232750</a> * <a href="/A291137" title="Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of inverse of k-th cyclotomic po...">A291137</a> <a href="/A285421" title="1-limiting word of the morphism 0->11, 1-> 011.">A285421</a> <a href="/A285431" title="Fixed point of the morphism 0->11, 1-> 110.">A285431</a></div> <div class=sectline>Adjacent sequences: <a href="/A080761" title="Positive numbers of the form y^2 - x^3, x and y >= 1.">A080761</a> <a href="/A080762" title="Positive numbers not of the form y^2 - x^3, x and y >= 1.">A080762</a> <a href="/A080763" title="Exchange 1's and 2's in the eta-sequence A006337.">A080763</a> * <a href="/A080765" title="Integers m such that m+1 divides lcm(1 through m).">A080765</a> <a href="/A080766" title="A unitary phi reciprocal amicable number: consider two different numbers a, b which satisfy the following equation for some ...">A080766</a> <a href="/A080767" title="A unitary phi reciprocal amicable number: consider two different numbers a, b which satisfy the following equation for some ...">A080767</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:Matthew_Vandermast">Matthew Vandermast</a>, Mar 25 2003</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 10 10:17 EDT 2025. 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