This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A326924 #51 Oct 23 2022 01:43:54
%S A326924 0,9,2,5,8,3,6,1,4,7,10,13,28,31,14,11,26,23,44,19,22,43,40,17,34,37,
%T A326924 18,15,32,29,52,25,46,21,76,47,50,27,12,33,16,39,20,45,24,51,48,77,
%U A326924 114,73,70,105,38,35,60,93,30,53,84,49,78,115,74,41,68,103,36,61,94,57,54,85,124,81
%N A326924 Squares visited by a knight on a spirally numbered board, moving always to the unvisited square closest to the origin.
%C A326924 "Closest to the origin" is meant in the sense of Euclidean distance, and in case of a tie, the square coming earliest on the spiral.
%C A326924 Differs from the original A316328 from a(34) = 76 on. See there for more information and other related sequences.
%C A326924 The knight gets trapped at the 22325th move at position (x,y) = (81, -18), from which it can't reach any unvisited square.
%C A326924 Sequence A326922 gives the distance^2 of the square number a(n) visited at move n. - _M. F. Hasler_, Oct 22 2019
%C A326924 From _M. F. Hasler_, Nov 04 2019: (Start)
%C A326924 When a(22325) = 25983 at (81, -18) is reached, at distance sqrt(6885) from the origin, the last unvisited square has number 13924, at (-59, 59), distance sqrt(6962) from the origin. This suggests that in an infinite extension (knight moves one step back if no unvisited square is available, cf. A323809) the knight might eventually visit every square. Can this be disproved by a counterexample of a square which will never be visited in the infinite extension? (In A328908 such a counterexample exists even before the knight gets stuck.)
%C A326924 The ratio a(n)/n oscillates between 0.5 and less than 1.7 for all n > 3000, even < 1.5 for all n > 14000, cf. graph of the sequence. What is the superior and inferior limit of this ratio, assuming the infinite extension beyond n = 22325?
%C A326924 (End)
%H A326924 M. F. Hasler, <a href="/A326924/b326924.txt">Table of n, a(n) for n = 0..22325</a>
%H A326924 M. F. Hasler, <a href="/wiki/Knight_tours">Knight tours</a>, OEIS wiki, Nov. 2019.
%H A326924 M. F. Hasler, <a href="/A326924/a326924.gif">Plot of knight tour A326924(0..22325), ending at (x,y) = (81,-18)</a>, Oct 20 2022.
%o A326924 (PARI) {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])
%Y A326924 Cf. A316328, A326918, A326922.
%Y A326924 Cf. A326413, A326916, A316667; A328908, A328928; A328909, A328929; A326413, A328698.
%Y A326924 Cf. A174344, A274923, A296030 (coordinate of square number n).
%K A326924 nonn,fini,full
%O A326924 0,2
%A A326924 _M. F. Hasler_, Oct 21 2019