A253268 -id:A253268 - cp's OEIS frontend

A253270 Decimal expansion of a constant related to A253268.

1, 1, 1, 8, 8, 6, 7, 4, 4, 3, 8, 6, 1, 7, 7, 8, 9, 5, 0, 1, 7, 3, 5, 3, 5, 5, 6, 9, 0, 5, 5, 7, 7, 1, 1, 0, 0, 5, 4, 9, 2, 6, 7, 2, 6, 4, 2, 4, 2, 2, 2, 3, 7, 4, 2, 8, 1, 5, 7, 1, 2, 2, 5, 3, 1, 3, 1, 8, 0, 1, 4, 3, 2, 9, 5, 7, 3, 2, 0, 6, 6, 5, 0, 3, 9, 0, 6, 7, 0, 8, 9, 6, 4, 8, 3, 0, 6, 8, 3, 0, 1, 1, 6, 6, 6

Offset: 1

Vaclav Kotesovec, May 01 2015

			1.1188674438617789501735355690557711005492672642422237428157122531318...

Cf. A062381, A253268.

(* Iterations: *) Do[Print[N[Product[Fibonacci[k]^k,{k,1,n}] / (GoldenRatio^(n*(n+1)*(2*n+1)/6) / 5^(n*(n+1)/4)),120]],{n,100,1000,100}]

Equals limit n->infinity A253268(n) / (((1+sqrt(5))/2)^(n*(n+1)*(2*n+1)/6) / 5^(n*(n+1)/4)).

A062381 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).

Original entry on oeis.org

1, -2, -360, 16848000, 1897448716800000, -3129723891582775706419200000, -541942196790147039091108680776954796441600000, 66373536294235576434745706427960099542896427384297349714149376000000

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 08 2001

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.

			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.

Colin Barker, Table of n, a(n) for n = 1..19
T. M. Richardson, The Filbert Matrix, arXiv:math/9905079 [math.RA], 1999.

Cf. A000045, A010048.
Cf. A003266, A152686, A253268.
Cf. A062073, A253267, A253270.

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 *)

vector(8, n, 1/matdet(matrix(n, n, i, j, 1/fibonacci(i+j-1)))) \\ Colin Barker, May 01 2015

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

a(n) ~ (-1)^floor(n/2) * A253270 * ((1+sqrt(5))/2)^(n*(2*n^2+1)/3) / (A253267^2 * 5^(n/2) * A062073^(2*n-2)). - Vaclav Kotesovec, May 01 2015

More terms from Vladeta Jovovic, Jul 11 2001

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A253270 Decimal expansion of a constant related to A253268.

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A062381 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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