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A003987 - 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>A003987 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A003987" 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=%2fA003987">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="A003987 - 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> A003987 </div> <div class=seqname> Table of n XOR m (or Nim-sum of n and m) read by antidiagonals with m&gt;=0, n&gt;=0. </div> </div> <div class=scorerefs> 213 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>0, 1, 1, 2, 0, 2, 3, 3, 3, 3, 4, 2, 0, 2, 4, 5, 5, 1, 1, 5, 5, 6, 4, 6, 0, 6, 4, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 6, 4, 6, 0, 6, 4, 6, 8, 9, 9, 5, 5, 1, 1, 5, 5, 9, 9, 10, 8, 10, 4, 2, 0, 2, 4, 10, 8, 10, 11, 11, 11, 11, 3, 3, 3, 3, 11, 11, 11, 11, 12, 10, 8, 10, 12, 2, 0, 2, 12, 10, 8, 10, 12, 13, 13, 9, 9, 13, 13, 1, 1, 13, 13, 9, 9, 13, 13</div> <div class=seqdatalinks> (<a href="/A003987/list">list</a>; <a href="/A003987/table">table</a>; <a href="/A003987/graph">graph</a>; <a href="/search?q=A003987+-id:A003987">refs</a>; <a href="/A003987/listen">listen</a>; <a href="/history?seq=A003987">history</a>; <a href="/search?q=id:A003987&fmt=text">text</a>; <a href="/A003987/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,4</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>Another way to construct the array: construct an infinite square matrix starting in the top left corner using the rule that each entry is the smallest nonnegative number that is not in the row to your left or in the column above you.</div> <div class=sectline>After a few moves the [symmetric] matrix looks like this:</div> <div class=sectline> 0 1 2 3 4 5 ...</div> <div class=sectline> 1 0 3 2 5 ...</div> <div class=sectline> 2 3 0 1 ?</div> <div class=sectline> 3 2 1</div> <div class=sectline> 4 5 ?</div> <div class=sectline> 5</div> <div class=sectline>The ? is then replaced with a 6.</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 60.</div> <div class=sectline>J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.</div> <div class=sectline>Eric Friedman, Scott M. Garrabrant, Ilona K. Phipps-Morgan, A. S. Landsberg and Urban Larsson, Geometric analysis of a generalized Wythoff game, in Games of no Chance 5, MSRI publ. Cambridge University Press, date?</div> <div class=sectline>D. Gale, Tracking the Automatic Ant and Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998; see p. 190. [From <a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a>, Jul 14 2009]</div> <div class=sectline>R. K. Guy, Impartial games, pp. 35-55 of Combinatorial Games, ed. R. K. Guy, Proc. Sympos. Appl. Math., 43, Amer. Math. Soc., 1991.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>T. D. Noe, <a href="/A003987/b003987.txt">Rows n = 0..100 of triangle, flattened</a></div> <div class=sectline>J.-P. Allouche and J. Shallit, <a href="https://doi.org/10.1016/S0304-3975(03)00090-2">The Ring of k-regular Sequences, II</a>, Theoret. Computer Sci., 307 (2003), 3-29.</div> <div class=sectline>R茅my Sigrist, <a href="/A003987/a003987.png">Colored representation of T(x,y) for x = 0..1023 and y = 0..1023</a> (where the hue is function of T(x,y) and black pixels correspond to zeros)</div> <div class=sectline>N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).</div> <div class=sectline>N. J. A. Sloane, <a href="https://arxiv.org/abs/2105.05111">The OEIS: A Fingerprint File for Mathematics</a>, arXiv:2105.05111 [math.HO], 2021. Mentions this sequence.</div> <div class=sectline><a href="/index/Ni#Nimsums">Index entries for sequences related to Nim-sums</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>T(2i,2j) = 2T(i,j), T(2i+1,2j) = 2T(i,j) + 1.</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>Table begins</div> <div class=sectline> 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...</div> <div class=sectline> 1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, ...</div> <div class=sectline> 2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8, ...</div> <div class=sectline> 3, 2, 1, 0, 7, 6, 5, 4, 11, 10, ...</div> <div class=sectline> 4, 5, 6, 7, 0, 1, 2, 3, 12, ...</div> <div class=sectline> 5, 4, 7, 6, 1, 0, 3, 2, ...</div> <div class=sectline> 6, 7, 4, 5, 2, 3, 0, ...</div> <div class=sectline> 7, 6, 5, 4, 3, 2, ...</div> <div class=sectline> 8, 9, 10, 11, 12, ...</div> <div class=sectline> 9, 8, 11, 10, ...</div> <div class=sectline> 10, 11, 8, ...</div> <div class=sectline> 11, 10, ...</div> <div class=sectline> 12, ...</div> <div class=sectline> ...</div> <div class=sectline>The first few antidiagonals are</div> <div class=sectline> 0;</div> <div class=sectline> 1, 1;</div> <div class=sectline> 2, 0, 2;</div> <div class=sectline> 3, 3, 3, 3;</div> <div class=sectline> 4, 2, 0, 2, 4;</div> <div class=sectline> 5, 5, 1, 1, 5, 5;</div> <div class=sectline> 6, 4, 6, 0, 6, 4, 6;</div> <div class=sectline> 7, 7, 7, 7, 7, 7, 7, 7;</div> <div class=sectline> 8, 6, 4, 6, 0, 6, 4, 6, 8;</div> <div class=sectline> 9, 9, 5, 5, 1, 1, 5, 5, 9, 9;</div> <div class=sectline> 10, 8, 10, 4, 2, 0, 2, 4, 10, 8, 10;</div> <div class=sectline> 11, 11, 11, 11, 3, 3, 3, 3, 11, 11, 11, 11;</div> <div class=sectline> 12, 10, 8, 10, 12, 2, 0, 2, 12, 10, 8, 10, 12;</div> <div class=sectline> ...</div> <div class=sectline>[Symmetric] matrix in base 2:</div> <div class=sectline> 0 1 10 11 100 101, 110 111 1000 1001 1010 1011 ...</div> <div class=sectline> 1 0 11 10 101 100, 111 110 1001 1000 1011 ...</div> <div class=sectline> 10 11 0 1 110 111, 100 101 1010 1011 ...</div> <div class=sectline> 11 10 1 0 111 110, 101 100 1011 ...</div> <div class=sectline> 100 101 110 111 0 1 10 11 ...</div> <div class=sectline> 101 100 111 110 1 0 11 ...</div> <div class=sectline> 110 111 100 101 10 11 ...</div> <div class=sectline> 111 110 101 100 11 ...</div> <div class=sectline> 1000 1001 1010 1011 ...</div> <div class=sectline> 1001 1000 1011 ...</div> <div class=sectline> 1010 1011 ...</div> <div class=sectline> 1011 ...</div> <div class=sectline> ...</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline>nimsum := proc(a, b) local t1, t2, t3, t4, l; t1 := convert(a+2^20, base, 2); t2 := convert(b+2^20, base, 2); t3 := evalm(t1+t2); map(x-&gt;x mod 2, t3); t4 := convert(evalm(%), list); l := convert(t4, base, 2, 10); sum(l[k]*10^(k-1), k=1..nops(l)); end; # memo: adjust 2^20 to be much bigger than a and b</div> <div class=sectline>AT := array(0..N, 0..N); for a from 0 to N do for b from a to N do AT[a, b] := nimsum(a, b); AT[b, a] := AT[a, b]; od: od:</div> <div class=sectline># alternative:</div> <div class=sectline>read(&quot;transforms&quot;) :</div> <div class=sectline><a href="/A003987" title="Table of n XOR m (or Nim-sum of n and m) read by antidiagonals with m&gt;=0, n&gt;=0.">A003987</a> := proc(n, m)</div> <div class=sectline> XORnos(n, m) ;</div> <div class=sectline>end proc: # <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Apr 17 2013</div> <div class=sectline>seq(seq(Bits:-Xor(k, m-k), k=0..m), m=0..20); # <a href="/wiki/User:Robert_Israel">Robert Israel</a>, Dec 31 2015</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Flatten[Table[BitXor[b, a - b], {a, 0, 10}, {b, 0, a}]] (* BitXor and Nim Sum are equivalent *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) tabl(nn) = {for(n=0, nn, for(k=0, n, print1(bitxor(k, n - k), &quot;, &quot;); ); print(); ); };</div> <div class=sectline>tabl(13) \\ <a href="/wiki/User:Indranil_Ghosh">Indranil Ghosh</a>, Mar 31 2017</div> <div class=sectline>(Python)</div> <div class=sectline>for n in range(14):</div> <div class=sectline> print([k^(n - k) for k in range(n + 1)]) # <a href="/wiki/User:Indranil_Ghosh">Indranil Ghosh</a>, Mar 31 2017</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Initial rows are <a href="/A001477" title="The nonnegative integers.">A001477</a>, <a href="/A004442" title="Natural numbers, pairs reversed: a(n) = n + (-1)^n; also Nimsum n + 1.">A004442</a>, <a href="/A004443" title="Nimsum n + 2.">A004443</a>, <a href="/A004444" title="Nimsum n + 3.">A004444</a>, etc. Cf. <a href="/A051775" title="Table T(n,m) = Nim-product of n and m, read by antidiagonals, for n &gt;= 0, m &gt;= 0.">A051775</a>, <a href="/A051776" title="Table T(n,m) = Nim-product of n and m, read by antidiagonals, for n &gt;= 1, m &gt;= 1.">A051776</a>.</div> <div class=sectline>Cf. <a href="/A003986" title="Table T(n,k) = n OR k read by antidiagonals.">A003986</a> (OR), <a href="/A004198" title="Table of x AND y, where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...">A004198</a> (AND), <a href="/A221146" title="Table read by antidiagonals: (m+n) - (m XOR n).">A221146</a> (carries).</div> <div class=sectline>Antidiagonal sums are in <a href="/A006582" title="a(n) = Sum_{k=1..n-1} k XOR n-k.">A006582</a>.</div> <div class=sectline>Sequence in context: <a href="/A332448" title="a(n) = A007814(A087808(sigma(n))).">A332448</a> <a href="/A184829" title="a(n) = smallest k such that A000961(n+1) = A000961(n) + (A000961(n) mod k), or 0 if no such k exists.">A184829</a> <a href="/A321132" title="a(n) is the number of iterations of the mapping of x -&gt; pi(x) until n reaches the main line as defined by A007097.">A321132</a> * <a href="/A307302" title="Array read by antidiagonals: Sprague-Grundy values for the game NimHof with rules [1,0], [3,3], [0,1].">A307302</a> <a href="/A307297" title="Array read by antidiagonals: Sprague-Grundy values for the game NimHof with 4 rules [1,0], [2,1], [3,3], [0,1],">A307297</a> <a href="/A307301" title="Array read by antidiagonals: Sprague-Grundy values for the game NimHof with rules [1,0], [3,1], [0,1].">A307301</a></div> <div class=sectline>Adjacent sequences: <a href="/A003984" title="Table of max(x,y), where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...">A003984</a> <a href="/A003985" title="Triangle with subscripts (1,1),(2,1),(1,2),(3,1),(2,2),(1,3), etc. in which entry (i,j) is i AND j.">A003985</a> <a href="/A003986" title="Table T(n,k) = n OR k read by antidiagonals.">A003986</a> * <a href="/A003988" title="Triangle with subscripts (1,1),(2,1),(1,2),(3,1),(2,2),(1,3), etc. in which entry (i,j) is [ i/j ].">A003988</a> <a href="/A003989" title="Triangle T from the array A(x, y) = gcd(x,y), for x &gt;= 1, y &gt;= 1, read by antidiagonals.">A003989</a> <a href="/A003990" title="Table of lcm(x,y), read along antidiagonals.">A003990</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="typically a triangle of numbers, made into a sequence by reading it row by row"><a href="/A003987/table">tabl</a></span>,<span title="a sequence of nonnegative numbers">nonn</span>,<span title="an exceptionally nice sequence">nice</span>,<span title="sequence has an interesting graph"><a href="/A003987/graph">look</a></span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Marc_LeBrun">Marc LeBrun</a></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 March 11 03:49 EDT 2025. Contains 381631 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>

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