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A094817
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Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 3.
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2
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2, 6, 19, 62, 206, 692, 2340, 7944, 27032, 92112, 314128, 1071776, 3657824, 12485696, 42623040, 145512576, 496787840, 1696093440, 5790732544, 19770612224, 67500721664, 230461137920, 786842059776, 2686443866112
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OFFSET
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1,1
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COMMENTS
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In general, a(n) = (2/m)*Sum_{r=1..m-1} sin(r*j*Pi/m)*sin(r*k*Pi/m)*(2*cos(r*Pi/m))^(2n) counts (s(0), s(1), ..., s(2n)) such that 0 < s(i) < m and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = j, s(2n) = k.
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LINKS
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FORMULA
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a(n) = (1/4) * Sum_{r=1..7} sin(3*r*Pi/8)^2*(2*cos(r*Pi/8))^(2*n).
a(n) = 6*a(n-1) - 10*a(n-2) + 4*a(n-3), n >= 4.
G.f.: -x*(2-6*x+3*x^2) / ( (2*x-1)*(2*x^2-4*x+1) ).
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MATHEMATICA
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Rest@ CoefficientList[Series[-x (2 - 6 x + 3 x^2)/((2 x - 1) (2 x^2 - 4 x + 1)), {x, 0, 24}], x] (* Michael De Vlieger, Feb 12 2022 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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