login
a(n) is the last square visited by fers moves on a spirally numbered (2n-1) X (2n-1) board, moving to the lowest available unvisited square at each step.
1

%I #34 Sep 08 2022 08:46:21

%S 1,3,15,29,61,87,139,177,249,299,391,453,565,639,771,857,1009,1107,

%T 1279,1389,1581,1703,1915,2049,2281,2427,2679,2837,3109,3279,3571,

%U 3753,4065,4259,4591,4797,5149,5367,5739,5969,6361,6603,7015,7269,7701,7967,8419

%N a(n) is the last square visited by fers moves on a spirally numbered (2n-1) X (2n-1) board, moving to the lowest available unvisited square at each step.

%C A 5 X 5 board, for example, is numbered with the square spiral:

%C .

%C 21--22--23--24--25

%C |

%C 20 7---8---9--10

%C | | |

%C 19 6 1---2 11

%C | | | |

%C 18 5---4---3 12

%C | |

%C 17--16--15--14--13

%C .

%C A fers is a (1,1)-leaper and can move one square diagonally.

%H Colin Barker, <a href="/A308385/b308385.txt">Table of n, a(n) for n = 1..1000</a>

%H Stephen Emmerson and Geoff Foster, <a href="http://www.theproblemist.org/dloads/Glossary.pdf">A glossary of fairy chess definitions</a>, British Chess Problem Society, 2018.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ferz">Ferz</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

%F a(n) = (4n^2-9n+6)*[n is odd] + (4n^2-11n+9)*[n is even] where [] is the Iverson bracket.

%F a(n) = A054556(n)*[n is odd] + (A054552(n)+1)*[n is even] where [] is the Iverson bracket.

%F a(n) = A316884(n^2)*[n is odd] + A316884(n^2-n)*[n is even] where [] is the Iverson bracket.

%F From _Colin Barker_, May 23 2019: (Start)

%F G.f.: x*(1 + 2*x + 10*x^2 + 10*x^3 + 9*x^4) / ((1 - x)^3*(1 + x)^2).

%F a(n) = (3/2)*(5+(-1)^n) - (10+(-1)^n)*n + 4*n^2.

%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.

%F (End)

%F E.g.f.: (1/2)*exp(-x)*(3 + 2*x + exp(2*x)*(15 - 12*x + 8*x^2)) - 9. - _Stefano Spezia_, Aug 17 2019

%t Table[(3/2) (5 + (-1)^n) - (10 + (-1)^n) n + 4 n^2, {n, 60}] (* _Vincenzo Librandi_, Aug 01 2019 *)

%o (PARI) Vec(x*(1 + 2*x + 10*x^2 + 10*x^3 + 9*x^4) / ((1 - x)^3*(1 + x)^2) + O(x^40)) \\ _Colin Barker_, May 23 2019

%o (Magma) [(3/2)*(5+(-1)^n) - (10+(-1)^n)*n + 4*n^2: n in [1..50]]; // _Vincenzo Librandi_, Aug 01 2019

%Y Cf. A054552, A054556, A316667, A316884.

%K nonn,easy

%O 1,2

%A _Sangeet Paul_, May 23 2019