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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
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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).

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

MATHEMATICA

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

PROG

(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

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

CROSSREFS

Cf. A054552, A054556, A316667, A316884.

Sequence in context: A018784 A147344 A201434 * A202506 A053519 A039666

Adjacent sequences:  A308382 A308383 A308384 * A308386 A308387 A308388

KEYWORD

nonn,easy

AUTHOR

Sangeet Paul, May 23 2019

STATUS

approved

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Last modified April 6 16:14 EDT 2020. Contains 333276 sequences. (Running on oeis4.)