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Number of non-attacking knights on an n X n board with all non-perimeteral squares removed.
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%I #17 Apr 18 2022 09:47:21

%S 1,4,4,8,12,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,80,84,

%T 88,92,96,100,104,108,112,116,120,124,128,132,136,140,144,148,152,156,

%U 160,164,168,172,176,180,184,188,192,196,200,204,208,212,216,220,224,228

%N Number of non-attacking knights on an n X n board with all non-perimeteral squares removed.

%C The basic maximal arrangement is with a knight in each corner and any centers of edges. The remaining 16 squares are coupled together, so only half may be used.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

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

%F G.f.: x*(x+1)*(4*x^5-8*x^4+8*x^3-4*x^2+x+1)/(x-1)^2. [_Colin Barker_, Jan 09 2013]

%e One of the maximal arrangements for a(8):

%e k-kkkk-k

%e -......-

%e k......k

%e k......k

%e k......k

%e k......k

%e -......-

%e k-kkkk-k

%o (PARI) a(n)=local(r); r=4+4*max(0,n-6); if (n>3, r+=8); if (n==4, r-=4); r for (i=2,40,print1(a(i)","))

%K nonn,easy

%O 1,2

%A _Jon Perry_, Jul 27 2003

%E More terms from _David Wasserman_, Mar 28 2005