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A326924 Squares visited by a knight on a spirally numbered board, moving always to the unvisited square closest to the origin. 13
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, 18, 15, 32, 29, 52, 25, 46, 21, 76, 47, 50, 27, 12, 33, 16, 39, 20, 45, 24, 51, 48, 77, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

"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.

Differs from the original A316328 from a(34) = 76 on. See there for more information and other related sequences.

The knight gets trapped at the 22325th move, where it can't reach any unvisited square.

Sequence A326922 gives the distance^2 of the square number a(n) visited at move n. - M. F. Hasler, Oct 22 2019

From M. F. Hasler, Nov 04 2019: (Start)

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.)

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?

(End)

LINKS

M. F. Hasler, Table of n, a(n) for n = 0..22325

M. F. Hasler, Knight tours, OEIS wiki, Nov. 2019.

PROG

(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])

CROSSREFS

Cf. A316328, A326918, A326922.

Cf. A326413, A326916, A316667; A328908, A328928; A328909, A328929; A326413, A328698.

Cf.  A174344, A274923, A296030 (coordinate of square number n).

Sequence in context: A316328 A323809 A328909 * A328908 A275967 A155799

Adjacent sequences:  A326921 A326922 A326923 * A326925 A326926 A326927

KEYWORD

nonn,fini,full

AUTHOR

M. F. Hasler, Oct 21 2019

STATUS

approved

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Last modified May 30 05:35 EDT 2020. Contains 334712 sequences. (Running on oeis4.)