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A316667(n)=A316328(n-1)+1 \\ M. F. Hasler, Nov 06 2019
The board is spirally numbered, starting with 0 at (0,0), as follows: . 16--15--14--13--12 : | | : 17 4---3---2 11 28 | | | | | 18 5 0---1 10 27 | | | | 19 6---7---8---9 26 | | 20--21--22--23--24--25 . Coordinates of a point are given in A174344, A274923 and A296030 (but these have offset 1: they list coordinates of the n-th point on the spiral, so the coordinates of first point, 0 at the origin, have index n = 1, etc). Starting at the origin, a(0) = 0, the knight jumps to the square with the lowest number at the eight available positions, (+-2, +-1) or (+-1, +-2), which is a(1) = 9 at (2, -1). From there, the available square with the lowest number is a(2) = 2 at (1, 1): square 0 at the origin is not available since already occupied earlier. Similarly, the knight will not be allowed to go on squares a(1) = 9 or a(2) = 2 ever after.
{local( K=[[(-1)^(i\2)<<(i>4),(-1)^i<<(i<5)]|i<-[1..8]], nxt(p, x=coords(p))=vecsort(apply(K->t(x+K), K))[1], pos(x,y)=if(y>=abs(x),4*y^2-y-x,-x>=abs(y),4*x^2-x-y,-y>=abs(x),(4*y-3)*y+x,(4*x-3)*x+y), coords(n, m=sqrtint(n), k=m\/2)=if(m<=n-=4*k^2, [n-3*k, -k], n>=0, [-k, k-n], n>=-m, [-k-n, k], [k, 3*k+n]), U=[], t(x, p=pos(x[1],x[2]))=if(p<=U[1]||setsearch(U, p), oo, p)); my(A=List(0)); for(n=1, oo, U=setunion(U, [A[n]]); while(#U>1&&U[2]==U[1]+1, U=U[^1]); iferr(listput(A, nxt(A[n])), E, break)); print("Index of the last term: ", #A-1); A316328(n)=A[n+1];}
The squares are numbered as in the spiral given in A174344 (upside down to get a counterclockwise spiral, but this is irrelevant here). The knight starts at a(0) = 0 with coordinates (0, 0). It jumps to a(1) = 9 with coordinates (2, -1): all 8 available squares (+-2, +-1) and (+-1, +-2) are at the same sup-norm and Euclidean distance from the origin, but square number 9 has the smallest number.
{local(coords(n, m=sqrtint(n), k=m\/2)=if(m<=n-=4*k^2, [n-3*k, -k], n>=0, [-k, k-n], n>=-m, [-k-n, k], [k, 3*k+n]), U=[]/*used squares*/, K=vector(8, i, [(-1)^(i\2)<<(i>4), (-1)^i<<(i<5)])/* knight moves*/, pos(x,y)=if(y>=abs(x), 4*y^2-y-x, -x>=abs(y),4*x^2-x-y, -y>=abs(x),(4*y-3)*y+x, (4*x-3)*x+y), t(x, p=pos(x[1],x[2]))=if(p<=U[1]||setsearch(U, p), oo, [vecmax(abs(x)), norml2(x), p]), nxt(p, x=coords(p))=vecsort(apply(K->t(x+K), K))[1][3]); my(A=List(0)/*list of positions*/); for(n=1, oo, U=setunion(U, [A[n]]); while(#U>1&&U[2]==U[1]+1, U=U[^1]); iferr(listput(A, nxt(A[n])), E, break)); print("Index of last term: ", #A-1); A328909(n)=A[n+1];} \\ To compute the infinite extension of the sequence, set en upper limit to the for() loop and replace "break" by listput(A, A[n-1])
{local(coords(n, m=sqrtint(n), k=m\/2)=if(m<=n-=4*k^2, [n-3*k, -k], n>=0, [-k, k-n], n>=-m, [-k-n, k], [k, 3*k+n]), U=[]/* used squares */, K=vector(8, i, [(-1)^(i\2)<<(i>4), (-1)^i<<(i<5)])/* knight moves */, pos(x,y)=if(y>=abs(x),4*y^2-y-x, -x>=abs(y),4*x^2-x-y, -y>=abs(x),(4*y-3)*y+x, (4*x-3)*x+y), t(x, p=pos(x[1],x[2]))=if(p<=U[1]||setsearch(U, p), oo, [norml2(x),p]), nxt(p, x=coords(p))=vecsort(apply(K->t(x+K), K))[1][2]); my(A=List(0)/*list of positions*/); for(n=1, oo, U=setunion(U, [A[n]]); while(#U>1&&U[2]==U[1]+1, U=U[^1]); iferr(listput(A, nxt(A[n])), E, break)); print("Index of last term: ", #A-1); A326924(n)=A[n+1];} \\ To compute the infinite extension, set upper bound in for() loop and replace "break" by listput(A, A[n-1])
The squares are numbered as in the spiral given in A174344 (upside down to get a counterclockwise spiral, but this is irrelevant here). The knight starts at a(0) = 0 with coordinates (0, 0). It jumps to a(1) = 9 with co-ordinates (2, -1): all 8 available squares (+-2, +-1) and (+-1, +-2) are at the same taxicab (2 + 1 = 3) and Euclidean distance (sqrt(2^2 + 1^2) = sqrt(5)) from the origin, but square number 9 has the smallest number. a(73) = 175 with coordinates (7, 0) is the first destination which is preferred due to the 1-norm (= 7) over A326924(73) = 81 with coordinates (5, -4), having 1-norm 5 + 4 = 9 but Euclidean or 2-norm sqrt(41) smaller than 7. a(1000) = 816 with coordinates (-10, -14). a(2000) = 2568 with coordinates (-7, -25). a(5000) = 4476 with coordinates (21, -33). a(10000) = 15560 with coordinates (-2, -62). a(20000) = 19566 with coordinates (-36, 70). a(50000) = 62092 with coordinates (125, -33). a(10^5) = 135634 with coordinates (-184, -26), taxicab distance 210 from the origin. a(200'000) = 259798 with coordinates (47, 255). a(500'000) = 713534 with coordinates (-68, -422). a(1'000'000) = 995288 with coordinates (217, 499). a(1'092'366) = 1165672 with coordinates (188, 540), taxicab norm 728 from the origin, is the last square visited by the knight before there is no unvisited square within reach. By then the earliest square on the spiral not yet visited is number 629641 at (397, 396), taxicab norm 793, and the unvisited square closest to the origin is number 1794929 at (1, 670), taxicab norm 671.
{Nmax=1e5;/* Full seq. with > 10^6 terms takes long to compute. */ local( K=[[(-1)^(i\2)<<(i>4),(-1)^i<<(i<5)]|i<-[1..8]], pos(x,y)=if(y>=abs(x),4*y^2-y-x,-x>=abs(y),4*x^2-x-y,-y>=abs(x),(4*y-3)*y+x,(4*x-3)*x+y), coords(n, m=sqrtint(n), k=m\/2)=if(m<=n-=4*k^2, [n-3*k, -k], n>=0, [-k, k-n], n>=-m, [-k-n, k], [k, 3*k+n]), t(x, p=pos(x[1],x[2]))=if(pt(x+K), K))[1][3], U=0,Umin=0); my(A=List(0)); for(n=1, Nmax, U+=1<<(A[n]-Umin); while(bittest(U,0), U>>=1;Umin++); iferr(listput(A, nxt(A[n])), E, break)); print("Index of the last term: ", #A-1); A328908(n)=A[n+1];}
The board is numbered with the square spiral: 17--16--15--14--13 : | | : 18 5---4---3 12 29 | | | | | 19 6 1---2 11 28 | | | | 20 7---8---9--10 27 | | 21--22--23--24--25--26 See A323809 for examples where "backtracking" happens. - _M. F. Hasler_, Nov 06 2019
A323808(n)=A323809(n-1)+1 \\ M. F. Hasler, Nov 06 2019
{L326916=List(0) /* list of terms */; U326916=1 /* bitmap of used squares */; local( K=vector(8, i, [(-1)^(i\2)<<(i>4), (-1)^i<<(i<5)])/* knight moves */, coords(n, m=sqrtint(n), k=m\/2)=if(m<=n-=4*k^2, [n-3*k, -k], n>=0, [-k, k-n], n>=-m, [-k-n, k], [k, 3*k+n]), pos(x, y)=if(y>=abs(x), 4*y^2-y-x, -x>=abs(y), 4*x^2-x-y, -y>=abs(x), (4*y-3)*y+x, (4*x-3)*x+y), val(x, p=pos(x[1],x[2]))=if(bittest(U326916, p), oo, [A007376(p), norml2(x), p])); iferr( for(n=1,oo, my(x=coords(L326916[n])); U326916+=1<A326916(n)=L326916[n+1]} \\ Requires function A007376; defines function A326916.
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