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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.

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A326916 Trajectory of the knight's tour for choice of the square with the lowest digit, then closest to the origin, then first in the spiral.

Original entry on oeis.org

0, 11, 14, 31, 28, 51, 10, 13, 34, 95, 190, 247, 312, 385, 244, 133, 242, 239, 376, 301, 372, 233, 370, 295, 232, 173, 228, 367, 230, 171, 226, 223, 358, 285, 220, 355, 282, 217, 352, 283, 218, 115, 44, 73, 20, 71, 40, 17, 36, 15, 18, 3, 12, 1, 22, 75, 46, 117, 48, 77, 24, 79, 50, 81, 118, 221, 286, 225, 292, 229, 296, 451, 298, 235
Offset: 0

Views

Author

M. F. Hasler, Oct 21 2019

Keywords

Comments

A variant of Angelini's "Kneil's Knumberphile Knight", inspired by Sloane's "The Trapped Knight", cf. A316328 and links:
Consider an infinite chess board with squares numbered along the infinite square spiral starting with 0 at the origin (as in A174344, A274923 and A296030). The squares are filled with successive digits of the integers: 0, 1, 2, ..., 9, 1, 0, 1, 1, ... (= A007376 starting with 0). The knight moves at each step to the yet unvisited square with the lowest digit on it, and in case of a tie, the one closest to the origin, first by Euclidean distance, then by appearance on the spiral (i.e., number of the square). This sequence lists the number of the square visited in the n-th move, if the knight starts at the origin, viz a(0) = 0.
It turns out that following these rules, the knight gets trapped at the 1070th move, when he can't reach any unvisited square.
See A326918 for the sequence of visited digits, given as A007376(a(n)).
Many squares, e.g., 2: (1,1), 4: (-1,1), 5: (-1,0), 6: (-1,-1), 7: (0,-1), 8: (1,-1), 9: (2,-1), ..., will never be visited, even in the infinite extension of the sequence where the knight can move back if it gets trapped, in order to resume with a new unvisited square, as in A323809. - M. F. Hasler, Nov 08 2019

Links

Crossrefs

Cf. A326918; A316328, A316667; A326924, A326922; A328908, A328928; A328909, A328929; A326413, A328698.
Cf. A174344, A274923, A296030; A007376.

Programs

  • PARI
    {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.

A326931 a(n) is the end square spiral number for a knight starting on square n moving on a board with squares numbered with the square of their distance from the 0-square origin and where the knight moves to the smallest numbered unvisited square; the smallest spiral number ordering is used if the distances are equal.

Original entry on oeis.org

25984, 51159, 8224, 31440, 8224, 31440, 8224, 110081, 131178, 92879, 69289, 59225, 62391, 10042, 66686, 73825, 36212, 123343, 158628, 28616, 74166, 98142, 59386, 50028, 42525, 15828, 7092, 27981, 57726, 27313, 52761, 15586, 47169, 17233, 152620, 73042, 76303, 83957, 59892, 9567
Offset: 1

Views

Author

Scott R. Shannon, Oct 22 2019

Keywords

Comments

This is the end square, using its spiral numbered value, for a knight starting on a square with spiral number n for a knight with step rules given in A326922. We use the spiral number to define the start and end square, as opposed to its square distance from the 0-square origin which predominantly determines the knight's path in A326922, as it is a unique value for each square on the board.
The largest end square spiral value for starting squares n from 1 to 200000 is a(72000) = 574108, which has a square distance number of 149725, which was also the largest found value. The largest number of steps before being trapped is for start square 103623, which is trapped after 483425 steps.
The smallest end square spiral value is a(1284) = 1143, which has a square distance number of 298. The smallest number of steps before being trapped is for start square 633, which is trapped after 1127 steps on square 1206. This has a square distance number of 293, the smallest value found.

Examples

			a(1) = 25984. See A326922.

Links

  • Scott R. Shannon, Table of n, a(n) for n = 1..20000
  • Scott R. Shannon, Path for starting square n = 72000. This has the largest found end square spiral number of 574108. In this and other images a green square marks the starting square, a red square the ending square, and blue squares mark the eight blocking squares for the end square. The end square is on the edge at about 12:30 on a clock.
  • Scott R. Shannon, Path for starting square n = 103623. This is trapped after 483425 steps, the largest found value. The end square is on the edge at about 6:30 on a clock.
  • Scott R. Shannon, Path for starting square n = 1284. This has the smallest found end square spiral number of 1143. Note that the start square acts as one of the eight blocking squares for the end square.
  • Scott R. Shannon, Path for starting square n = 633. This is trapped after 1127 steps, the smallest found value.
  • N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019).

Crossrefs

Cf. A316667, A309918, A326922, A326924.

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.

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Oct 19 2022

Keywords

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

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]]
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