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 1-3 of 3 results.

A324275 Numbers k for which A324274(k) is 0, i.e., starting squares in A324274 that yield a path of infinite length.

Original entry on oeis.org

2, 9, 14, 27, 34, 53, 64, 89, 102, 133, 150, 187, 206, 249, 272, 321, 346, 401, 430, 491, 522, 589, 624, 697, 734, 813, 854, 939, 982, 1073, 1120, 1217, 1266, 1369, 1422, 1531, 1586, 1701, 1760, 1881, 1942, 2069, 2134, 2267, 2334, 2473
Offset: 1

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Author

Jan Koornstra, Feb 27 2019

Keywords

Comments

Note that the sequence up to a(n) (for its current known values) is actually the path of a(n) in reverse until it reaches square 2. It is therefore conjectured that all starting squares in A324274 either have a finite length or are part of this single sequence.

Crossrefs

Formula

Conjectures from Colin Barker, Mar 09 2019: (Start)
G.f.: x*(2 + 7*x + 3*x^2 + 6*x^3 - x^5 + x^6) / ((1 - x)^3*(1 + x)^2*(1 + x^2)).
a(n) = (5 + 7*(-1)^n + (2-2*i)*(-i)^n + (2+2*i)*i^n + (26+6*(-1)^n)*n + 18*n^2) / 16 where i=sqrt(-1).
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7) for n>7.
(End)

A324273 Squares visited by a single pawn move for an even square and a double pawn move for an odd square on a diagonally numbered board and moving to the lowest available unvisited square of different parity at each step.

Original entry on oeis.org

1, 4, 7, 2, 5, 12, 17, 8, 13, 6, 3, 10, 15, 26, 19, 32, 25, 14, 9, 18
Offset: 1

Views

Author

Jan Koornstra, Feb 20 2019

Keywords

Comments

The 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 .
.

Examples

			Square 1 is odd. Hence the next square should be the lowest even square a double move away from 1, which is square 4. Next, there is only a single option to move in a single move to an odd square, namely at square 7. Etc.
		

Crossrefs

A307344 Cells visited by a single pawn move for an even cell and a double pawn move for an odd cell on a numbered 3D grid and moving to the lowest available unvisited cell of different parity at each step.

Original entry on oeis.org

1, 10, 19, 48, 27, 76, 51, 20, 33, 8, 15, 4, 9, 34, 53, 108, 77, 28, 13, 2, 5, 30, 47, 18, 31, 6, 3, 12, 23, 72, 49, 102, 71, 22, 11, 36, 21, 70, 101, 186, 131, 252, 193, 106, 75, 26, 43, 16
Offset: 1

Views

Author

Jan Koornstra, Apr 02 2019

Keywords

Comments

The grid is numbered as follows:
1: [0, 0, 0]
-- 1 step --
2: [0, 0, 1]
3: [0, 1, 0]
4: [1, 0, 0]
-- 2 steps --
5: [0, 0, 2]
6: [0, 1, 1]
7: [0, 2, 0]
8: [1, 0, 1]
9: [1, 1, 0]
10: [2, 0, 0]
etc.

Examples

			1: [0, 0, 0] is an odd cell, hence a double move is required. Since 5: [0, 0, 2] and 7: [0, 2, 0] are also odd, 10: [2, 0, 0] is the only valid move.
The sequence ends at 16: [1, 1, 1]. A single move is required, which limits the possible destination cells to:
   6: [0, 1, 1], even;
   8: [1, 0, 1], even;
   9: [1, 1, 0], already visited;
  27: [1, 1, 2], already visited;
  28: [1, 2, 1], even;
  31: [2, 1, 1], already visited;
		

Crossrefs

Showing 1-3 of 3 results.