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
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 17 2018
STATUS
approved