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a(n) = L(n)*a(n-1) + a(n-2) with a(0) = a(1) = 1 and L(n) the Lucas numbers A000032.
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%I #21 Nov 19 2020 22:08:49

%S 1,1,4,17,123,1370,24783,720077,33868402,2574718629,316724259769,

%T 63030702412660,20296202901136289,10574384742194419229,

%U 8914226633872796546336,12159015702987236683621533,26834956570719465233549269667,95827642073054913336241125602390

%N a(n) = L(n)*a(n-1) + a(n-2) with a(0) = a(1) = 1 and L(n) the Lucas numbers A000032.

%C Denominators of the continued fraction expansion [2, 1, 3, . . . , L(n)], where L(n) represents the n-th Lucas Number.

%C 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)]).

%C 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

%F a(n) ~ c * ((1 + sqrt(5))/2)^(n*(n+1)/2), where c = 2.8051534321074771176277443455334066418353792262447...

%p a:= proc(n) option remember; `if`(n<2, 1, a(n-1)*

%p (<<0|1>, <1|1>>^n. <<2, 1>>)[1, 1]+a(n-2))

%p end:

%p seq(a(n), n=0..20); # _Alois P. Heinz_, Nov 19 2020

%t Table[Denominator[FromContinuedFraction[LucasL[Range[0, n]]]], {n, 0, 20}]

%Y Cf. A000032, A135829.

%K nonn

%O 0,3

%A _Greg Dresden_, Aug 30 2020