A302999 a(n) = Product_{k=1..n} (Fibonacci(k+2) - 1).

1, 1, 2, 8, 56, 672, 13440, 443520, 23950080, 2107607040, 301387806720, 69921971159040, 26290661155799040, 16011012643881615360, 15786858466867272744960, 25195826113120167300956160, 65080818850189392138369761280, 272037822793791659138385602150400

Offset

0,3

Comments

a(n) = determinant of (n + 1) X (n + 1) matrix whose main diagonal consists of the consecutive Fibonacci numbers starting with Fibonacci(2) (1, 2, 3, 5, 8, 13, ...) and all other elements are 1's (see example).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..97

Formula

a(n) = Product_{k=1..n} A000071(k+2).

a(n) = Product_{k=1..n} Sum_{j=1..k} A000045(j).

a(n) ~ c * ((1 + sqrt(5))/2)^(n*(n+5)/2) / 5^(n/2), where c = 0.1972502311584232476952451740107000852343536766534965116633336539193... - Vaclav Kotesovec, Apr 17 2018

a(n) = A190535(n-3) for n > 3. - Alois P. Heinz, Apr 25 2018

Example

The matrix begins:
  1  1  1  1  1  1  1  1  ...
  1  2  1  1  1  1  1  1  ...
  1  1  3  1  1  1  1  1  ...
  1  1  1  5  1  1  1  1  ...
  1  1  1  1  8  1  1  1  ...
  1  1  1  1  1 13  1  1  ...
  1  1  1  1  1  1 21  1  ...
  1  1  1  1  1  1  1 34  ...

Maple

b:= proc(n) b(n):= `if`(n<1, [1$2][], (f->
      [f, b(n-1)[2]*(f-1)][])(b(n-1)+b(n-2)))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 24 2018

Mathematica

Table[Product[Fibonacci[k + 2] - 1, {k, 1, n}], {n, 0, 17}]
Table[Product[Sum[Fibonacci[j], {j, 1, k}], {k, 1, n}], {n, 0, 17}]
Table[Det[Table[If[i == j, Fibonacci[i + 1], 1], {i, 1, n + 1}, {j, 1, n + 1}]], {n, 0, 17}]

Crossrefs

Cf. A000045, A000071, A003266, A067962, A190535, A194157, A194158, A196870.

Sequence in context: A108208 A203199 A348875 * A135079 A084872 A254231

Adjacent sequences: A302996 A302997 A302998 * A303000 A303001 A303002

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 17 2018

Status

approved