A357046 Squares visited by a knight moving on a board covered with horizontal dominoes [m|m], m = 0, 1, 2, ... in a diamond-shaped spiral, when the knight always jumps to the unvisited square with the least number on the corresponding domino.

0, 11, 14, 1, 4, 13, 10, 3, 18, 7, 2, 5, 22, 9, 28, 31, 60, 15, 32, 29, 52, 25, 8, 27, 12, 53, 26, 23, 6, 17, 34, 59, 30, 87, 126, 51, 24, 45, 20, 39, 16, 33, 58, 55, 86, 125, 50, 47, 76, 21, 40, 67, 36, 61, 94, 57, 54, 85, 176, 129, 56, 93, 138, 187, 92, 137, 96, 35, 38, 19

Offset: 0

Author

M. F. Hasler, Oct 19 2022

Keywords

• nonn
• fini
• full

Comments

The sequence lists the squares visited by the knight by giving their (unique) "square spiral number", as shown, e.g., in A316328 and others. (Listing the labels m of the dominoes would obviously be ambiguous; see EXAMPLE for that sequence.)

The dominoes [m|m], m = 0, 1, 2, ... are placed in a diamond-shaped spiral,

12 12 28 28

_ 13 13 11 11 27 27 _

14 14 [2 | 2] 10 10 26 26

_ 15 15 [3 | 3] [1 | 1] [9 | 9] 25

_ 16 [4 | 4] [0 | 0] [8 | 8] 24 24

17 17 [5 | 5] [7 | 7] 23 23

_ 18 18 [6 | 6] 22 22 _

19 19 21 21

The spiral starts from the origin (where the [0|0] is placed) with one step in direction North-East (where [1|1] is placed), then one in direction North-West (=> [2|2]), then two towards South-West (=> [3|3] and [4|4]) and two towards South-East (=> [5|5] and [6|6]), then three towards North-East, etc. [We chose the counter-clockwise spiral as usual in mathematics, but one would obviously get the same sequence if the spiral of dominoes and the square spiral numbering the positions were chosen in the opposite, clockwise sense.]

The endpoints of the "straight lines" are labeled with the "quarter-squares" A002620, in particular, rightmost and leftmost dominoes of each "shell" are labeled with the odd resp. even square numbers.

The sequence ends at a(2550) where the knight is stuck at position (x, y) = (28, 4) on the domino labeled m = 964.

Examples

			The knight hops from the left 0 (= the origin) on the right 1, then on the left 2, then on the right 0, then on the left 3, then on the right 2, etc.
The list of these labels would be 0, 1, 2, 0, 3, 2, 8, 3, 4, 5, 1, 4, 6, 7, 9, 11, 12, 14, 11, 10, 24, 22, 7, 8, 10, 9, 23, 6, 5, 15, 13, 12, 27, 26, 48, 23, ...
As explained in comments, the terms a(n) correspond to the (unique) "square spiral numbers" of these locations (cf. A274641 or A174344 (upside down) or A316328).
		

Links

• Eric Angelini, Spirals for Scott, Blog entry, Oct. 19, 2022.
• Eric Angelini, Spirals for Scott, Blog entry, Oct. 19, 2022 [Local copy, with permission].
• M. F. Hasler, Plot of the knight's tour A357046, Oct 20 2022.

Crossrefs

Cf. A316328, A326924 and A326922 (choose square closest to the origin), A328908 and A328928 (variant using taxicab distance); A328909 and A328929 (variant using sup norm).

Cf. A274641, A174344 (upside down), A268038, A274923 for the square spiral numbering and corresponding (x,y) coordinates.

Programs

PARI

/* function domino([x,y]) gives the label m on the domino at (x,y); it uses the map DOM to store this label with key x + i*y. */
DOM=Map(); {domino(x)=while(!mapisdefined(DOM, x[1]+I*x[2], &x), my(M=#DOM\2, side=sqrtint(M*4-!!M), pos=sqrtint(M)*I^(side-1)+side\/2%2*I, dir=(1+I)*I^side); for(m=M, M+side\2, mapput(DOM, pos, m); mapput(DOM, pos+1, m); pos+=dir)); x}
{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])}
{local(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, [domino(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); A357046(n)=A[n+1];} \\ same code as A326924 except for norml2 => domino
/* to get the sequence of labels m (cf.example): */
[domino(coords(A357046(n))) | n <- [0..99]]