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Numbers appearing on the upper face of a die as a result of its turning over the edge while it rolls along the square spiral of natural numbers.
2

%I #23 Mar 05 2023 12:08:08

%S 1,2,3,1,4,2,3,1,4,6,2,1,5,4,2,3,5,1,2,6,5,3,2,4,5,3,6,4,1,3,6,5,1,2,

%T 6,5,1,3,6,4,1,3,6,5,1,2,6,5,1,2,3,5,4,2,3,5,4,1,3,6,4,1,3,6,4,5,3,2,

%U 4,5,3,2,4,6,3,1,4,6,3,1,4,6,2,1,5,6,2,1,5,6,2,4,5,3,2,4

%N Numbers appearing on the upper face of a die as a result of its turning over the edge while it rolls along the square spiral of natural numbers.

%C Along the lines of the grid of an infinite checkered plane, a spiral one cell wide is drawn, twisting clockwise (see the figure at the link). A die with numbers 1, 2, 3, 4, 5 and 6, in which the sum of points on opposite sides is 7, is placed in the initial cell of the spiral so that 1 is located on its upper side, 4 on the front, and 5 on the right. The size of the face of the cube coincides with the size of the cell of the plane. The cube, rolling over the edge, gets into the next cell in a spiral, and so on, moving along the cells of the drawn spiral. Each cell of the spiral contains a number located on the upper face of the game cube that rolled along it, and thus the sequence is obtained.

%H Nicolay Avilov, <a href="/A361136/a361136.jpg">Rolling a die in a square spiral</a>

%e This is how the first 56 terms of the sequence are obtained:

%e .

%e 43--44--45--46--47--48--49--50 6---5---1---2---6---5---1---2

%e | | | |

%e 42 21--22--23--24--25--26 51 3 5---3---2---4---5---3 3

%e | | | | | | | |

%e 41 20 7---8---9--10 27 52 1 6 3---1---4---6 6 5

%e | | | | | | | | | | | |

%e 40 19 6 1---2 11 28 53 4 2 2 1---2 2 4 4

%e | | | | | | | | | | | | | |

%e 39 18 5---4---3 12 29 54 6 1 4---1---3 1 1 2

%e | | | | | | | | | |

%e 38 17--16--15--14--13 30 55 3 5---3---2---4---5 3 3

%e | | | | | |

%e 37--36--35--34--33--32--31 56 1---5---6---2---1---5---6 5

%e .

%e Comparing the spirals, we get, for example: a(10) = 6, a(25) = 5, a(37) = 1.

%K nonn

%O 1,2

%A _Nicolay Avilov_, Mar 02 2023