A308385 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, 3, 15, 29, 61, 87, 139, 177, 249, 299, 391, 453, 565, 639, 771, 857, 1009, 1107, 1279, 1389, 1581, 1703, 1915, 2049, 2281, 2427, 2679, 2837, 3109, 3279, 3571, 3753, 4065, 4259, 4591, 4797, 5149, 5367, 5739, 5969, 6361, 6603, 7015, 7269, 7701, 7967, 8419

Offset: 1

Author

Sangeet Paul, May 23 2019

Keywords

• nonn
• easy

Comments

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

.

21--22--23--24--25

|

20 7---8---9--10

| | |

19 6 1---2 11

| | | |

18 5---4---3 12

| |

17--16--15--14--13

.

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

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Stephen Emmerson and Geoff Foster, A glossary of fairy chess definitions, British Chess Problem Society, 2018.
• Wikipedia, Ferz
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A054552, A054556, A316667, A316884.

Programs

Magma

[(3/2)*(5+(-1)^n) - (10+(-1)^n)*n + 4*n^2: n in [1..50]]; // Vincenzo Librandi, Aug 01 2019

Mathematica

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

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

Formula

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

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

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

From Colin Barker, May 23 2019: (Start)

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

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

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

(End)

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