cp's OEIS Frontend

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.

Showing 1-10 of 25 results. Next

A316334 Same as A316588, except number the squares starting at 0 rather than 1.

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, 70, 81, 67
Offset: 0

Views

Author

N. J. A. Sloane, Jul 14 2018

Keywords

Comments

a(n) = A316588(n+1)-1.
See A316588 for further information.

Links

Crossrefs

Cf. A316588, A316336.

A316335 Numbers missing from A316588.

Original entry on oeis.org

1596, 1651, 1652, 1653, 1707, 1708, 1709, 1710, 1711, 1764, 1765, 1766, 1767, 1768, 1769, 1770, 1822, 1823, 1824, 1825, 1826, 1827, 1828, 1829, 1830, 1881, 1882, 1883, 1884, 1885, 1886, 1887, 1888, 1889, 1890, 1891, 1941, 1942, 1943, 1944, 1945, 1946, 1947, 1948, 1949, 1950, 1951, 1952, 1953, 2001
Offset: 1

Views

Author

N. J. A. Sloane, Jul 14 2018

Keywords

Comments

A316588 is finite, so this sequence is infinite.
See A316588 for further information.

Crossrefs

Cf. A316588, A316334, A316336.

A316667 Squares visited by a knight moving on a spirally numbered board always to the lowest available unvisited square.

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
Offset: 1

Views

Author

Daniël Karssen, Jul 10 2018, following a suggestion from N. J. A. Sloane, Jul 09 2018

Keywords

Comments

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
.
This sequence is finite: At step 2016, square 2084 is visited, after which there are no unvisited squares within one knight move.

Links

Crossrefs

Cf. A316328 (same starting at 0), A329022 (same with diamond-shaped spiral), A316588 (variant on board with x,y >= 0).
Cf. A326924 (choose square closest to the origin), A328908 (using taxicab distance), A328909 (using sup norm); A323808, A323809.
The (x,y) coordinates of square k are (A174344(k), A274923(k)).

Programs

Formula

a(n) = A316328(n-1) + 1.

A316328 Lexicographically earliest knight's path on spiral on infinite chessboard.

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
Offset: 0

Views

Author

N. J. A. Sloane, Jul 13 2018

Keywords

Comments

On a doubly-infinite chessboard, number all the cells in a counterclockwise spiral starting at a central cell labeled 0. Start with a knight at cell 0, and thereafter always move the knight to the smallest unvisited cell. Sequence gives succession of squares visited.
Sequence ends if knight is unable to move.
Inspired by A316588 and, like that sequence, has only finitely many terms; see A316667 for details.
See A326924 for a variant where the knight prefers squares closest to the origin, and gets trapped only after 22325 moves. - M. F. Hasler, Oct 21 2019
See A323809 for an infinite extension of this sequence, obtained by allowing the knight to go back in case it was trapped. See A328908 for a variant of length > 10^6, using the taxicab distance, and A328909 for a variant using the sup norm. - M. F. Hasler, Nov 04 2019

Examples

			The board is spirally numbered, starting with 0 at (0,0), as follows:
.
  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
.
Coordinates of a point are given in A174344, A274923 and A296030 (but these have offset 1: they list coordinates of the n-th point on the spiral, so the coordinates of first point, 0 at the origin, have index n = 1, etc).
Starting at the origin, a(0) = 0, the knight jumps to the square with the lowest number at the eight available positions, (+-2, +-1) or (+-1, +-2), which is a(1) = 9 at (2, -1).
From there, the available square with the lowest number is a(2) = 2 at (1, 1): square 0 at the origin is not available since already occupied earlier. Similarly, the knight will not be allowed to go on squares a(1) = 9 or a(2) = 2 ever after.

Links

Crossrefs

Cf. A316667 (same with offset 1 and values +1), A316338 (numbers not in this sequence).
Cf. A323809 (infinite extension of this sequence).
Cf. A316588 (variant with diagonally numbered board, coordinates x, y >= 0).
Cf. A326924 and A326922 (variant: choose square closest to the origin), A328908 and A328928 (variant using taxicab distance); A328909 and A328929 (variant using sup norm).
Cf. A326916 and A326918, A326413, A328698 (squares are filled with digits of the infinite word 0,1,...9,1,0,1,1,...).
Cf. A174344, A274923, A296030 (coordinates of a given square).

Programs

  • PARI
    {local( K=[[(-1)^(i\2)<<(i>4),(-1)^i<<(i<5)]|i<-[1..8]], nxt(p, x=coords(p))=vecsort(apply(K->t(x+K), K))[1], 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=[], t(x, p=pos(x[1],x[2]))=if(p<=U[1]||setsearch(U, p), oo, p)); my(A=List(0)); 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 the last term: ", #A-1); A316328(n)=A[n+1];}

Formula

a(n) = A316667(n+1) - 1.

Extensions

Terms from a(17) on computed by Daniël Karssen, Jul 10 2018
Examples added and crossrefs edited by M. F. Hasler, Nov 04 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

Views

Author

Daniël Karssen, Jan 28 2019

Keywords

Comments

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

Examples

			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

Links

Crossrefs

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).

Programs

Formula

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

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

Views

Author

Daniël Karssen, Jan 28 2019

Keywords

Comments

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)

Examples

			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)

Links

Crossrefs

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).

Programs

  • PARI
    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];}

Formula

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

Extensions

Edited by M. F. Hasler, Nov 02 2019

A316671 Squares visited by moving diagonally one square on a diagonally numbered board and moving to the lowest available unvisited square at each step.

Original entry on oeis.org

1, 5, 4, 12, 11, 23, 22, 38, 37, 57, 56, 80, 79, 107, 106, 138, 137, 173, 172, 212, 211, 255, 254, 302, 301, 353, 352, 408, 407, 467, 466, 530, 529, 597, 596, 668, 667, 743, 742, 822, 821, 905, 904, 992, 991, 1083, 1082, 1178, 1177, 1277, 1276, 1380, 1379
Offset: 1

Views

Author

Daniël Karssen, Jul 15 2018

Keywords

Comments

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 .
.

Links

Crossrefs

Cf. A316588, A316668, A316669, A316670.

Programs

  • Mathematica
    CoefficientList[ Series[-(2x^4 - 3x^2 + 4x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 52}], x] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {1, 5, 4, 12, 11}, 53] (* Robert G. Wilson v, Jul 18 2018 *)
  • PARI
    Vec(x*(1 + 4*x - 3*x^2 + 2*x^4) / ((1 - x)^3*(1 + x)^2) + O(x^40)) \\ Colin Barker, Jul 18 2018

Formula

From Colin Barker, Jul 18 2018: (Start)
G.f.: x*(1 + 4*x - 3*x^2 + 2*x^4) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5.
a(n) = (n^2 + n + 4)/2 for n even.
a(n) = (n^2 - n + 2)/2 for n odd.
(End)

A316668 Squares visited by king moves on a diagonally numbered board and moving to the lowest available unvisited square at each step.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 8, 9, 6, 10, 14, 13, 12, 11, 16, 17, 18, 19, 20, 15, 21, 27, 26, 25, 24, 23, 22, 29, 30, 31, 32, 33, 34, 35, 28, 36, 44, 43, 42, 41, 40, 39, 38, 37, 46, 47, 48, 49, 50, 51, 52, 53, 54, 45, 55, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 67, 68
Offset: 1

Views

Author

Daniël Karssen, Jul 15 2018

Keywords

Comments

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 .
.
Same as A316588 but with king move instead of knight move.

Links

Crossrefs

Cf. A316588, A316669, A316670, A316671.

A316669 Squares visited by queen moves on a diagonally numbered board and moving to the lowest available unvisited square at each step.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 9, 7, 8, 10, 14, 11, 12, 13, 15, 20, 16, 17, 18, 19, 21, 27, 22, 23, 24, 25, 26, 28, 35, 29, 30, 31, 32, 33, 34, 36, 44, 37, 38, 39, 40, 41, 42, 43, 45, 54, 46, 47, 48, 49, 50, 51, 52, 53, 55, 65, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 77
Offset: 1

Views

Author

Daniël Karssen, Jul 15 2018

Keywords

Comments

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 .
.
Same as A316588 but with queen move instead of knight move.

Links

Crossrefs

Cf. A316588, A316668, A316670, A316671.

A316670 Squares visited by bishop moves on a diagonally numbered board and moving to the lowest available unvisited square at each step.

Original entry on oeis.org

1, 5, 4, 6, 14, 11, 12, 13, 15, 27, 22, 23, 24, 25, 26, 28, 44, 37, 38, 39, 40, 41, 42, 43, 45, 65, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 90, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 119, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117
Offset: 1

Views

Author

Daniël Karssen, Jul 15 2018

Keywords

Comments

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 .
.
Same as A316588 but with bishop move instead of knight move.

Links

Crossrefs

Cf. A316588, A316668, A316669, A316671.
Showing 1-10 of 25 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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