cp's OEIS Frontend

A363370 Number of ways to distribute n guards on the corners and walls of a square castle so that each wall has an equal number of guards modulo rotations and reflections.

%I A363370 #21 Jun 24 2023 12:48:50
%S A363370 1,0,1,1,3,2,5,4,9,8,13,13,22,20,30,31,44,44,60,61,82,84,106,111,141,
%T A363370 144,177,186,225,234,279,291,345,360,417,438,508,528,604,634,720,752,
%U A363370 848,886,996,1040,1156,1210,1345,1400,1545,1615,1775,1850,2025,2110
%N A363370 Number of ways to distribute n guards on the corners and walls of a square castle so that each wall has an equal number of guards modulo rotations and reflections.
%C A363370 The four walls of the castle are defended by n guards, each of whom is assigned to one of eight locations in the castle: one of the four towers at the corners of the castle, or the middle of one of the four walls. Each guard in a tower will defend the two adjoining walls, while each guard positioned at the middle of a wall will guard only that wall. The guards must be distributed so that each wall is defended by the same number of guards. If two distributions can be obtained from one another by reflection and/or rotation, they are counted as one.
%H A363370 Martins Opmanis, <a href="/A363370/b363370.txt">Table of n, a(n) for n = 0..1000</a>
%H A363370 <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 1, 1, -2, 1, -3, 1, 0, -1, 3, -1, 2, -1, -1, 0, -1, 1).
%F A363370 G.f.: (1/((1 - x^6)*(1 - x^4)) + 1/((1 - x^3)*(1 - x^2))^2)/(2*(1 - x^4)). - _Andrew Howroyd_, May 29 2023
%e A363370 For n = 5 there are two distinct distributions:
%e A363370   2-0-0    1-1-0
%e A363370   |   |    |   |
%e A363370   0   1    0   1
%e A363370   |   |    |   |
%e A363370   0-1-1    1-0-1
%o A363370 (PARI) Vec((1/((1 - x^6)*(1 - x^4)) + 1/((1 - x^3)*(1 - x^2))^2)/(2*(1 - x^4)) + O(x^61)) \\ _Andrew Howroyd_, May 29 2023
%K A363370 nonn
%O A363370 0,5
%A A363370 _Martins Opmanis_, May 29 2023
%E A363370 Terms a(22) and beyond from _Andrew Howroyd_, May 29 2023