A179611 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1+2*x)/(1 - 2*x - 8*x^2 - 4*x^3).
1, 4, 16, 68, 280, 1168, 4848, 20160, 83776, 348224, 1447296, 6015488, 25002240, 103917568, 431915008, 1795179520, 7461349376, 31011794944, 128895102976, 535729963008, 2226667929600, 9254755975168, 38465775239168
Offset: 0
Links
- Index entries for linear recurrences with constant coefficients, signature (2,8,4).
Programs
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Maple
with(LinearAlgebra): nmax:=22; m:=5; A[1]:= [0,1,0,1,1,0,0,0,0]: A[2]:= [1,0,1,1,1,1,0,0,0]: A[3]:= [0,1,0,0,1,1,0,0,0]: A[4]:=[1,1,0,0,1,0,1,1,0]: A[5]:= [0,0,0,0,0,1,1,1,1]: A[6]:= [0,1,1,0,1,0,0,1,1]: A[7]:= [0,0,0,1,1,0,0,1,0]: A[8]:= [0,0,0,1,1,1,1,0,1]: A[9]:= [0,0,0,0,1,1,0,1,0]: A:=Matrix([A[1],A[2],A[3],A[4],A[5],A[6],A[7],A[8],A[9]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);
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Mathematica
LinearRecurrence[{2,8,4},{1,4,16},30] (* Harvey P. Dale, Oct 20 2017 *)
Formula
G.f.: (1+2*x)/(1 - 2*x - 8*x^2 - 4*x^3).
a(n) = 2*a(n-1) + 8*a(n-2) + 4*a(n-3) with a(1)=1, a(2)=4 and a(3)=16.
a(n) = (8 + 3*z1 - 6*z1^2)*z1^(-n)/(z1*37) + (8 + 3*z2 - 6*z2^2)*z2^(-n)/(z2*37) + (8 + 3*z3 - 6*z3^2)*z3^(-n)/(z3*37) with z1, z2 and z3 the roots of f(x) = 1 - 2*x - 8*x^2 - 4*x^3 = 0.
alpha = arctan(3*sqrt(111));
z1 = sqrt(10)*cos(alpha/3)/6 + sqrt(30)*sin(alpha/3)/6 - 2/3 = 0.2405971520460078;
z2 = -sqrt(10)*cos(alpha/3)/3 - 2/3 = -1.585043243313016;
z3 = sqrt(10)*cos(alpha/3)/6 - sqrt(30)*sin(alpha/3)/6 - 2/3 = -0.6555539087329909.
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