OFFSET
0,3
COMMENTS
Denominators of the continued fraction expansion [2, 1, 3, . . . , L(n)], where L(n) represents the n-th Lucas Number.
The determinant of n X n matrix: ([L(1), -1, 0, . . . , 0], [1, L(2), -1, 0, . . . , 0], [0, 1, L(3), -1, 0, . . . , 0 ], . . . , [0, 0, 0, 0, . . . , 1, L(n)]).
Examples: a(1) = det[1] = 1; a(2) = det([[1, -1], [1, 3]]) = 4;a(3) = det([[1, -1, 0], [1, 3, -1], [0, 1, 4]]) = 17;a(4) = det([[1, -1, 0, 0], [1, 3 , -1, 0], [0, 1, 4, -1], [0, 0, 1, 7]]) = 123 - Seung Ju Lee, Sep 06 2020
FORMULA
a(n) ~ c * ((1 + sqrt(5))/2)^(n*(n+1)/2), where c = 2.8051534321074771176277443455334066418353792262447...
MAPLE
a:= proc(n) option remember; `if`(n<2, 1, a(n-1)*
(<<0|1>, <1|1>>^n. <<2, 1>>)[1, 1]+a(n-2))
end:
seq(a(n), n=0..20); # Alois P. Heinz, Nov 19 2020
MATHEMATICA
Table[Denominator[FromContinuedFraction[LucasL[Range[0, n]]]], {n, 0, 20}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Greg Dresden, Aug 30 2020
STATUS
approved