Daniël Karssen - cp's OEIS frontend

A323815 Squares where A323811 gets stuck.

Original entry on oeis.org

1377, 2849, 4464, 6785, 8910, 9729, 15752, 18914, 32639, 37127, 40185, 57629

Offset: 1

Daniël Karssen, Jan 28 2019

nonn

Cf. A323811.

A323814 Squares where A323810 gets stuck.

Original entry on oeis.org

1378, 2850, 4465, 6786, 8911, 9730, 15753, 18915, 32640, 37128, 40186, 57630

Offset: 1

Daniël Karssen, Jan 28 2019

nonn

Cf. A323810.

A323813 Squares where A323809 gets stuck.

Original entry on oeis.org

2083, 2719, 3324, 3752, 7775, 5631, 7410, 8561, 14075, 8468, 9230, 22701, 14660, 21709, 21077, 25808, 27111, 24707, 19843, 26942, 26736, 32448, 31365, 45035, 37852, 37187, 43317, 62094, 67400, 68735, 70847, 62788, 63222, 69244, 85384

Offset: 1

Daniël Karssen, Jan 28 2019

nonn

Daniël Karssen, Table of n, a(n) for n = 1..57

Cf. A323809.

A323811 Squares visited by a knight on a diagonally numbered board and moving to the lowest available unvisited square at each step and if no unvisited squares are available move one step back.

Original entry on oeis.org

0, 7, 5, 1, 11, 8, 3, 2, 12, 6, 4, 9, 25, 17, 10, 29, 23, 15, 37, 30, 21, 16, 24, 19, 27, 33, 13, 20, 42, 32, 26, 18, 14, 34, 41, 31, 22, 28, 38, 46, 55, 68, 36, 47, 39, 50, 59, 69, 56, 66, 80, 45, 57, 48, 40, 51, 43, 54, 63, 35, 64, 52, 44, 75, 62, 53, 65, 102, 87, 73, 60

Offset: 0

Daniël Karssen, Jan 28 2019

nonn

Board is numbered as follows:
   0  1  3  6 12 17  .
   2  4  7 13 18  .
   5 10 14 19  .
  11 15 20  .
  16 21  .
  22  .
  .
Coincides with A316334 for the first 2402 terms.

Daniël Karssen, Table of n, a(n) for n = 0..99999
Daniël Karssen, Figure showing the first 1e5 steps of the sequence

The sequences involved in this set of related sequences are A316588, A316328, A316334, A316667, A323808, A323809, A323810, and A323811.

A323810 Squares visited by a knight on a diagonally numbered board and moving to the lowest available unvisited square at each step and if no unvisited squares are available move one step back.

Original entry on oeis.org

1, 8, 6, 2, 12, 9, 4, 3, 13, 7, 5, 10, 26, 18, 11, 30, 24, 16, 38, 31, 22, 17, 25, 20, 28, 34, 14, 21, 43, 33, 27, 19, 15, 35, 42, 32, 23, 29, 39, 47, 56, 69, 37, 48, 40, 51, 60, 70, 57, 67, 81, 46, 58, 49, 41, 52, 44, 55, 64, 36, 65, 53, 45, 76, 63, 54, 66, 103, 88, 74, 61

Offset: 1

Daniël Karssen, Jan 28 2019

nonn

Board is numbered as follows:
   1  2  4  7 11 16  .
   3  5  8 12 17  .
   6  9 13 18  .
  10 14 19  .
  15 20  .
  21  .
  .
Coincides with A316588 for the first 2402 terms. - Daniël Karssen, Jan 30 2019

Daniël Karssen, Table of n, a(n) for n = 1..100000
Daniël Karssen, Figure showing the first 1e5 steps of the sequence

The sequences involved in this set of related sequences are A316588, A316328, A316334, A316667, A323808, A323809, A323810, and A323811.

A323809 Squares visited by a knight on a spirally numbered board, moving always to the lowest available unvisited square, or one step back if no unvisited square is available.

Original entry on oeis.org

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, 42, 69, 20, 39, 16, 33, 12, 27, 24, 45, 74, 41, 68, 103, 36, 61, 94, 57, 54, 85, 50, 47, 76, 113, 72, 107, 150, 67, 102, 63, 66, 35, 38, 65, 62

Offset: 0

Daniël Karssen, Jan 28 2019

This is an infinite extension of A316328, with which it coincides for the first 2016 terms. - N. J. A. Sloane, Jan 29 2019
From M. F. Hasler, Nov 04 2019: (Start)
At move 99999, the least yet unvisited square has number 66048, which is near the border of the visited region. This suggests that the knight will eventually visit every square. Can this be proved or disproved through a counterexample?
More formally, let us call "isolated" a set of unvisited squares which is connected through knight moves, but which cannot be extended to a larger such set by adding a further square. Can there exist at some moment a finite isolated set which the knight cannot reach? (Without the last condition, the square a(2016) would clearly satisfy the condition just before the knight reaches it.)
Such subsets have a good chance of preserving this property forever. It should be possible to prove that an isolated subset sufficiently far (2 knight moves?) from any other unvisited square (or from the infinite connected subset of unvisited squares) remains so forever. (This condition of distance could replace the time-dependent condition "reachable by the knight".)
If such (forever) isolated sets do exist, with what frequency will they occur? Could they have a nonzero asymptotic density? Will this (if so, how) depend on the way the knight measures "lowest available" (cf. variants where the taxicab or Euclidean or sup norm distance from the origin is used)? (End)

			The board is numbered following a square spiral:
  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
.
From _M. F. Hasler_, Nov 06 2019: (Start)
At move 2015, the knight lands on a(2015) = 2083, from where no unvisited squares can be reached. So the knight moves back to a(2016) = a(2014) = 2466, from where it goes on to the unvisited square a(2017) = 2667.
Similarly, at moves 2985, 3120, 3378, 7493, 8785, 9738, 10985, 11861, 11936, 12160, 18499, 18730, 19947 and 22251, the knight get "trapped" and has to move to the previous square on the next move.
On move 23044, the same happens on square 25808, and the knight must move back to square a(23045) = a(23043) = 27111. However, there is still no unvisited square in reach, so the knight has to make another step back to a(23046) = a(23042) = 28446, before it can move on to a(23047) = 29123. (End)

Daniël Karssen, Table of n, a(n) for n = 0..99999
M. F. Hasler, Knight tours, OEIS wiki, Nov. 2019.
Daniël Karssen, Figure showing the first 1e5 steps of the sequence

The sequences involved in this set of related sequences are A316588, A316328, A316334, A316667, A323808, A323809, A323810 and A323811.

Cf. A326924 & A326922 (using L2-norm), A328908 & A328928 (L1-norm), A328909 & A328929 (sup norm); A326916 & A326918 (digits on spiral), A326413 and A328698 (variants with other tie breaker).

Nmax=1e5 /* number of terms 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]), U=0, Umin=0, t(x, p=pos(x[1],x[2]))=if(pt(x+K), K))[1], back=0); my(A=List(0)); for(n=1, Nmax, back||U+=1<<(A[n]-Umin); while(bittest(U,0), U>>=1; Umin++); listput(A, nxt(A[n])); if(A[n+1] != oo, back=0, A[n+1]=A[n+1-back+=2])); print("Index of the last term: ", #A-1); A323809(n)=A[n+1];}

a(n) = A323808(n+1) - 1. - M. F. Hasler, Nov 06 2019

Edited by M. F. Hasler, Nov 02 2019

A323808 Squares visited by a knight on a spirally numbered board and moving to the lowest available unvisited square at each step and if no unvisited squares are available move one step back.

Original entry on oeis.org

1, 10, 3, 6, 9, 4, 7, 2, 5, 8, 11, 14, 29, 32, 15, 12, 27, 24, 45, 20, 23, 44, 41, 18, 35, 38, 19, 16, 33, 30, 53, 26, 47, 22, 43, 70, 21, 40, 17, 34, 13, 28, 25, 46, 75, 42, 69, 104, 37, 62, 95, 58, 55, 86, 51, 48, 77, 114, 73, 108, 151, 68, 103, 64, 67, 36, 39, 66, 63

Offset: 1

Daniël Karssen, Jan 28 2019

This is an infinite extension of A316667 with which it agrees for the first 2016 terms. - N. J. A. Sloane, Jan 28 2019

			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

Daniël Karssen, Table of n, a(n) for n = 1..100000
M. F. Hasler, Knight tours, OEIS wiki, Nov. 2019.
Daniël Karssen, Figure showing the first 1e5 steps of the sequence

The sequences involved in this set of related sequences are A316588, A316328, A316334, A316667, A323808, A323809, A323810, and A323811.

Cf. A326924 & A326922 (using L2-norm), A328908 & A328928 (L1-norm), A328909 & A328929 (sup norm); A326916 & A326918 (digits on spiral), A326413 and A328698 (variants with other tie breaker).

A323808(n)=A323809(n-1)+1 \\ M. F. Hasler, Nov 06 2019

a(n) = A323809(n-1) + 1. - M. F. Hasler, Nov 06 2019

A317918 Numbers missing from A317916.

Original entry on oeis.org

5329, 5398, 5399, 5402, 5403, 5405, 5407, 5429, 5613, 5622, 5626, 5667, 5679, 5698, 5702, 5704, 5705, 5764, 5840, 5850, 5882, 5922, 5924, 5927, 5930, 5937, 5940, 5943, 5975, 5985, 5992, 5995, 5996, 5999, 6000, 6001, 6003, 6004, 6005

Offset: 1

Daniël Karssen, Aug 11 2018

nonn

A317916 is finite, so this sequence is infinite.

See A317916 for further information.

Daniël Karssen, Table of n, a(n) for n = 1..10000

Cf. A317916.

A317917 Numbers missing from A317915.

Original entry on oeis.org

5328, 5397, 5398, 5401, 5402, 5404, 5406, 5428, 5612, 5621, 5625, 5666, 5678, 5697, 5701, 5703, 5704, 5763, 5839, 5849, 5881, 5921, 5923, 5926, 5929, 5936, 5939, 5942, 5974, 5984, 5991, 5994, 5995, 5998, 5999, 6000, 6002, 6003, 6004

Offset: 1

Daniël Karssen, Aug 11 2018

nonn

A317915 is finite, so this sequence is infinite.

See A317915 for further information.

Daniël Karssen, Table of n, a(n) for n = 1..10000

Cf. A317915.

A317916 Squares visited by a (2,5)-leaper on a spirally numbered board and moving to the lowest available unvisited square at each step, squares labelled >=1.

Original entry on oeis.org

1, 84, 31, 18, 27, 44, 15, 48, 41, 12, 37, 22, 33, 26, 19, 30, 47, 16, 43, 10, 39, 32, 23, 36, 29, 20, 49, 14, 45, 38, 11, 42, 35, 24, 69, 98, 187, 88, 61, 104, 193, 94, 53, 118, 173, 286, 163, 82, 177, 134, 249, 144, 203, 110, 345, 216, 71, 2, 67, 60, 7, 50, 13, 40

Offset: 1

Daniël Karssen, Aug 11 2018

Board is numbered with the square spiral:
  17--16--15--14--13
   |               |
  18   5---4---3  12   .
   |   |       |   |
  19   6   1---2  11   .
   |   |           |
  20   7---8---9--10   .
   |
  21--22--23--24--25--26
The sequence is finite: at step 7574, square 6760 is visited, after which there are no unvisited squares within one move.

Daniël Karssen, Table of n, a(n) for n = 1..7574
Daniël Karssen, Figure showing the first 35 steps of the sequence
Daniël Karssen, Figure showing the complete sequence

Cf. A317915, A317918.

a(n) = A317915(n-1) + 1.

cp's OEIS Frontend

User: Daniël Karssen

A323815 Squares where A323811 gets stuck.

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A323814 Squares where A323810 gets stuck.

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A323813 Squares where A323809 gets stuck.

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A323811 Squares visited by a knight on a diagonally numbered board and moving to the lowest available unvisited square at each step and if no unvisited squares are available move one step back.

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A323810 Squares visited by a knight on a diagonally numbered board and moving to the lowest available unvisited square at each step and if no unvisited squares are available move one step back.

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A323809 Squares visited by a knight on a spirally numbered board, moving always to the lowest available unvisited square, or one step back if no unvisited square is available.

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A323808 Squares visited by a knight on a spirally numbered board and moving to the lowest available unvisited square at each step and if no unvisited squares are available move one step back.

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A317918 Numbers missing from A317916.

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A317917 Numbers missing from A317915.

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A317916 Squares visited by a (2,5)-leaper on a spirally numbered board and moving to the lowest available unvisited square at each step, squares labelled >=1.

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