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 A068608 Path of a knight's tour on an infinite chessboard. 11
 1, 10, 3, 16, 19, 22, 9, 12, 15, 18, 7, 24, 11, 14, 5, 20, 23, 2, 13, 4, 17, 6, 21, 8, 25, 50, 27, 54, 31, 60, 35, 64, 67, 40, 71, 74, 45, 78, 49, 52, 29, 56, 59, 34, 63, 66, 39, 70, 43, 76, 47, 80, 51, 28, 55, 58, 33, 62, 37, 68 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS One of eight possible knight's tours. Squares are numbered in a clockwise spiral. Enumerates all positive integers. A description of the method to construct the tour is provided in A306659. - Hugo Pfoertner, May 11 2019 LINKS Hugo Pfoertner, Table of n, a(n) for n = 0..1088 Frank Ellermann, Sequences based on a spiral of the natural numbers Hugo Pfoertner, Illustration of tours corresponding to A068608-A068615, (2019). Dan Thomasson, Knight's Tour Art, (2001-2014). PROG (PARI) \\Ellermann's clockwise square spiral, first step (0, 0) -> (0, 1) y=vector(10000); L=0; d=1; n=0; for(r=1, 100, d=-d; k=floor(r/2)*d; for(j=1, L++, y[n++]=k); forstep(j=k-d, -floor((r+1)/2)*d+d, -d, y[n++]=j)); x=vector(10100); L=1; d=-1; n=0; for(r=1, 100, d=-d; k=floor(r/2)*d; for(j=1, L++, x[n++]=-k); forstep(j=k-d, -floor((r+1)/2)*d+d, -d, x[n++]=-j)); \\ Position in spiral findpos(i, j)={my(size=(2*max(abs(i), abs(j))+1)^2); forstep(k=size, 1, -1, if(i==x[k]&&j==y[k], return(k)))}; atan2(y, x)=if(x>0, atan(y/x), if(x==0, if(y>0, Pi/2, -Pi/2), if(y>=0, atan(y/x)+Pi, atan(y/x)-Pi))); angle(v, w)=atan2(v[1]*w[2]-v[2]*w[1], v[1]*w[1]+v[2]*w[2]); move=[2, 1; 1, 2; -1, 2; -2, 1; -2, -1; -1, -2; 1, -2; 2, -1]; \\ 8 Knight moves m=6; b=matrix(2*m+1, 2*m+1, i, j, 0); setb(pos)={b[pos[1]+m+1, pos[2]+m+1]=1}; getb(pos)=b[pos[1]+m+1, pos[2]+m+1]; inring(n, p)=!(abs(p[1])angmin, jmin=j; angmin=adiff; jlast=j))))); if(jmin>0, p+=move[jmin, ]; setb(p); ); ); p+=move[jlast, ]; setb(p)); \\ Hugo Pfoertner, May 11 2019 CROSSREFS Cf. A068609, A068610, A068611, A068612, A068613, A068614, A068615, A306659, A306660. Sequence in context: A309382 A064211 A050133 * A358278 A195817 A347126 Adjacent sequences: A068605 A068606 A068607 * A068609 A068610 A068611 KEYWORD nonn,look AUTHOR Hans Secelle and Albrecht Heeffer (albrecht.heeffer(AT)pandora.be), Mar 09 2002 STATUS approved

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Last modified December 5 05:37 EST 2023. Contains 367575 sequences. (Running on oeis4.)