A337521 a(n) = L(n)*a(n-1) + a(n-2) with a(0) = a(1) = 1 and L(n) the Lucas numbers A000032.

1, 1, 4, 17, 123, 1370, 24783, 720077, 33868402, 2574718629, 316724259769, 63030702412660, 20296202901136289, 10574384742194419229, 8914226633872796546336, 12159015702987236683621533, 26834956570719465233549269667, 95827642073054913336241125602390

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

Links

Table of n, a(n) for n=0..17.

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

Cf. A000032, A135829.

Sequence in context: A054927 A307794 A071138 * A032325 A032335 A208803

Adjacent sequences: A337518 A337519 A337520 * A337522 A337523 A337524

Keyword

nonn

Author

Greg Dresden, Aug 30 2020

Status

approved