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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.

%I A308385 #34 Sep 08 2022 08:46:21
%S A308385 1,3,15,29,61,87,139,177,249,299,391,453,565,639,771,857,1009,1107,
%T A308385 1279,1389,1581,1703,1915,2049,2281,2427,2679,2837,3109,3279,3571,
%U A308385 3753,4065,4259,4591,4797,5149,5367,5739,5969,6361,6603,7015,7269,7701,7967,8419
%N 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.
%C A308385 A 5 X 5 board, for example, is numbered with the square spiral:
%C A308385 .
%C A308385   21--22--23--24--25
%C A308385    |
%C A308385   20   7---8---9--10
%C A308385    |   |           |
%C A308385   19   6   1---2  11
%C A308385    |   |       |   |
%C A308385   18   5---4---3  12
%C A308385    |               |
%C A308385   17--16--15--14--13
%C A308385 .
%C A308385 A fers is a (1,1)-leaper and can move one square diagonally.
%H A308385 Colin Barker, <a href="/A308385/b308385.txt">Table of n, a(n) for n = 1..1000</a>
%H A308385 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 A308385 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ferz">Ferz</a>
%H A308385 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A308385 a(n) = (4n^2-9n+6)*[n is odd] + (4n^2-11n+9)*[n is even] where [] is the Iverson bracket.
%F A308385 a(n) = A054556(n)*[n is odd] + (A054552(n)+1)*[n is even] where [] is the Iverson bracket.
%F A308385 a(n) = A316884(n^2)*[n is odd] + A316884(n^2-n)*[n is even] where [] is the Iverson bracket.
%F A308385 From _Colin Barker_, May 23 2019: (Start)
%F A308385 G.f.: x*(1 + 2*x + 10*x^2 + 10*x^3 + 9*x^4) / ((1 - x)^3*(1 + x)^2).
%F A308385 a(n) = (3/2)*(5+(-1)^n) - (10+(-1)^n)*n + 4*n^2.
%F A308385 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
%F A308385 (End)
%F A308385 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 A308385 Table[(3/2) (5 + (-1)^n) - (10 + (-1)^n) n + 4 n^2, {n, 60}] (* _Vincenzo Librandi_, Aug 01 2019 *)
%o A308385 (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 A308385 (Magma) [(3/2)*(5+(-1)^n) - (10+(-1)^n)*n + 4*n^2: n in [1..50]]; // _Vincenzo Librandi_, Aug 01 2019
%Y A308385 Cf. A054552, A054556, A316667, A316884.
%K A308385 nonn,easy
%O A308385 1,2
%A A308385 _Sangeet Paul_, May 23 2019