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A062381
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Let A_n be the n X n matrix defined by A_n[i,j] = 1/F(i+j-1) for 1<=i,j<=n where F(k) is the k-th Fibonacci number (A000045). Then a_n = 1/det(A_n).
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14
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1, -2, -360, 16848000, 1897448716800000, -3129723891582775706419200000, -541942196790147039091108680776954796441600000, 66373536294235576434745706427960099542896427384297349714149376000000
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OFFSET
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1,2
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COMMENTS
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In the reference it is proved that not only det(A_n) is a reciprocal of an integer but the inverse matrix (A_n)^(-1) is an integer matrix.
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LINKS
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FORMULA
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a(n) = s(n) * f(n) / h(n)^2, where s(n) = (-1)^Floor[n/2], f(n) = Product[Fibonacci[k]^(n-Abs[k-n]),{k,1,2*n-1}], h(n) = Product[Product[Fibonacci[k],{k,1,m-1}],{m,1,n}]. - Alexander Adamchuk, May 18 2006
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EXAMPLE
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a(3) = -360 because the matrix is / 1,1,1/2 / 1,1/2, 1/3 / 1/2, 1/3, 1/5 / with determinant -1/360.
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MATHEMATICA
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Table[(-1)^Floor[n/2]*Product[Fibonacci[k]^(n-Abs[k-n]), {k, 1, 2*n-1}], {n, 1, 10}]/Table[Product[Product[Fibonacci[k], {k, 1, m-1}], {m, 1, n}], {n, 1, 10}]^2 (* Alexander Adamchuk, May 18 2006 *)
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PROG
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(PARI) vector(8, n, 1/matdet(matrix(n, n, i, j, 1/fibonacci(i+j-1)))) \\ Colin Barker, May 01 2015
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CROSSREFS
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KEYWORD
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sign,nice
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AUTHOR
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Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 08 2001
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EXTENSIONS
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STATUS
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approved
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