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A189145
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Number of n X 2 array permutations with each element making zero or one knight moves.
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19
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1, 1, 4, 16, 36, 81, 225, 625, 1600, 4096, 10816, 28561, 74529, 194481, 509796, 1336336, 3496900, 9150625, 23961025, 62742241, 164249856, 429981696, 1125736704, 2947295521, 7716041281, 20200652641, 52886200900, 138458410000
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OFFSET
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1,3
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COMMENTS
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a(n+2) is number of ways to place k non-attacking knights on a 2 x n board, sum over all k>=0.
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LINKS
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FORMULA
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Empirical: a(n) = 3*a(n-1) -3*a(n-2) +6*a(n-3) -6*a(n-5) +3*a(n-6) -3*a(n-7) +a(n-8).
Empirical: G.f. -x*(1-2*x+4*x^2+x^3+3*x^5+x^7-6*x^4-3*x^6) / ( (x-1)*(1+x)*(x^2-3*x+1)*(x^4+3*x^2+1) ). - R. J. Mathar, Oct 15 2011
Explicit formula: ((3+sqrt(5))/2)^(n+2)/25 + ((3-sqrt(5))/2)^(n+2)/25 + (((sqrt(5)+1)/2)^(n+2) + ((sqrt(5)-1)/2)^(n+2))*4*cos((Pi*n)/2)/25 + (((sqrt(5)+1)/2)^(n+2) - ((sqrt(5)-1)/2)^(n+2))*2*sin((Pi*n)/2)/25 + 1/10 + 7/50*(-1)^n. - Vaclav Kotesovec, Nov 07 2011
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EXAMPLE
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All solutions for 3X2
..0..1....0..4....5..1....5..4
..2..3....2..3....2..3....2..3
..4..5....1..5....4..0....1..0
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MATHEMATICA
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Table[FullSimplify[LucasL[2n+4]/25 + (3*Fibonacci[n+1] + Fibonacci[n]) * (2*Cos[(Pi*n)/2] + Sin[(Pi*n)/2])*2/25 + 7*(-1)^n/50 + 1/10], {n, 1, 20}] (* Vaclav Kotesovec, Nov 07 2011 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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