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A027994
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a(n) = (F(2n+3) - F(n))/2 where F() = Fibonacci numbers A000045.
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7
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1, 2, 6, 16, 43, 114, 301, 792, 2080, 5456, 14301, 37468, 98137, 256998, 672946, 1761984, 4613239, 12078110, 31621701, 82787980, 216743836, 567446112, 1485598681, 3889356696, 10182482353, 26658108074, 69791870526, 182717549872
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OFFSET
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0,2
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COMMENTS
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Substituting x*(1-x)/(1-2x) into x^2/(1-x^2) yields x^2*(g.f. of sequence).
The number of (s(0), s(1), ..., s(n+1)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n+1, s(0) = 2, s(n+1) = 3. - Herbert Kociemba, Jun 02 2004
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} T(n, k)*T(n, n+k), T given by A027926.
a(n) = 2*a(n-1) + Sum_{m < n-1} a(m) + F(n-1) = A059512(n+2) - F(n) where F(n) is the n-th Fibonacci number (A000045). - Floor van Lamoen, Jan 21 2001
a(n) = (2/5)*Sum_{k=1..4} sin(2*Pi*k/5)*sin(3*Pi*k/5)*(1+2*cos(Pi*k/5))^(n+1)). - Herbert Kociemba, Jun 02 2004
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MATHEMATICA
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Table[(Fibonacci[2n+3]-Fibonacci[n])/2, {n, 0, 30}] (* or *) LinearRecurrence[{4, -3, -2, 1}, {1, 2, 6, 16}, 30] (* Harvey P. Dale, Apr 28 2022 *)
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PROG
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(PARI) a(n)=(fibonacci(2*n+3)-fibonacci(n))/2
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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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