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 A189145 Number of n X 2 array permutations with each element making zero or one knight moves. 19
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Column 2 of A189150. a(n+2) is number of ways to place k non-attacking knights on a 2 x n board, sum over all k>=0. LINKS R. H. Hardin, Table of n, a(n) for n = 1..200 FORMULA 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 EXAMPLE 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 MATHEMATICA 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 *) CROSSREFS Sequence in context: A207170 A207069 A207436 * A005722 A075408 A206981 Adjacent sequences: A189142 A189143 A189144 * A189146 A189147 A189148 KEYWORD nonn AUTHOR R. H. Hardin, Apr 17 2011 STATUS approved

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Last modified September 29 02:45 EDT 2023. Contains 365749 sequences. (Running on oeis4.)