User sidharth iyer - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T09:54:23Z http://mathoverflow.net/feeds/user/16830 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90050/self-avoiding-walk-on-mathbbz Self-avoiding walk on \$\mathbb{Z}\$ Sidharth Iyer 2012-03-02T15:49:57Z 2012-03-04T08:32:04Z <p>This one is an <a href="http://math.stackexchange.com/questions/111377/self-avoiding-walk-on-mathbbz1.-" rel="nofollow">unanswered question</a> in Math.SE. I've posted it here because I think it deserves more attention.</p> <blockquote> <p>How many sequences \$\{a_n\}\$ exist satisfying: <p><em>a)</em> \$a_1=0\$ <p><em>b)</em> \$\forall k\ge1 \$ either \$a_{k+1}=a_k+k\$ or \$a_{k+1}=a_k-k\$ <p><em>c)</em> \$a_i \neq a_j\$ whenever \$i \neq j\$ <p><em>d)</em> \$\mathbb{Z}=\{a_i\}_{i > 0}\$</p> </blockquote> <p>Are the two below alternating sequences the only solutions?</p> <ul> <li>\$a_{2k}=k\$, \$a_{2k+1}=-k\$</li> <li>\$a_{2k}=-k\$, \$a_{2k+1}=k\$</li> </ul> <p>Also, it is known that <a href="http://mathworld.wolfram.com/RecamansSequence.html" rel="nofollow">Recamán's</a> <a href="https://oeis.org/A005132" rel="nofollow">sequence</a> satisfies <p><em>a)</em> \$a_1=1\$ and <em>b)</em>, <em>c)</em> as above </p> http://mathoverflow.net/questions/90050/self-avoiding-walk-on-mathbbz Comment by Sidharth Iyer Sidharth Iyer 2012-03-02T17:04:35Z 2012-03-02T17:04:35Z @MartinBrandenburg Thank you for the help! I've changed the title.