OFFSET
1,4
COMMENTS
From Thomas Scheuerle, Jan 13 2026: (Start)
We consider the sequence s(n) = 1,-1,2,-4,6,-8,12,-14,... with the ordinary generating function G(x) given as continued fraction expansion G(x) = 1/(1+F(1)*x^F(1)/(1+F(2)*x^F(2)/(1+F(3)/(...)))), where F(n) means the n-th Fibonacci number, then a(n+1) is the Hankel determinant over the first 2*F(n)-1 values of s(n).
a(10) = -1*2^215*3^86*5^50*7^34*13^42*17^21. (End)
LINKS
Paul Barry, On the Hankel transform of C-fractions, arXiv preprint arxiv:1212.3490 [math.CA], 2012.
FORMULA
From Thomas Scheuerle, Jan 13 2026: (Start)
If the generating function of a sequence is given by a continued fraction expansion of the form 1/(1-b*x/(1-c*x/(1-d*x/(1-e*x/(1-f*x/(1-g*x/(...))))))), then its Hankel transform is given by the sequence: 1, c*b, e*d*c^2*b^2, g*f*e^2*d^2*c^3*b^3, ... . This allows us to find a general formula:
|a(n)| = |a(n-1)|*( Product_{k=1..n-1} F(k) )^F(n-2), for n>2. The sign of the Hankel determinants has period 4, this means a(n) has negative sign if F(n)+1 is divisible by 4. (End)
PROG
(PARI)
b(n) = if(n<1, 1, b(n-1)*prod(k=1, n-1, fibonacci(k))^fibonacci(n-2))
a(n) = if((fibonacci(n)+1)%4==0, -b(n), b(n)) \\ Thomas Scheuerle, Jan 12 2026
CROSSREFS
KEYWORD
sign
AUTHOR
N. J. A. Sloane, Jan 25 2013
EXTENSIONS
More terms and name modified by Thomas Scheuerle, Jan 12 2026
STATUS
approved