A196870 a(n+1) = A001610(n)*a(n).

1, 2, 6, 36, 360, 6120, 171360, 7882560, 591192000, 72125424000, 14280833952000, 4584147698592000, 2383756803267840000, 2007123228351521280000, 2735708960243123504640000, 6034973966296330451235840000, 21544857059677899710911948800000

Offset: 1

Author

John M. Campbell, Oct 06 2011

Keywords

• nonn
• easy

Comments

Determinant of the n X n matrix with consecutive Lucas numbers along the main diagonal, and 1's everywhere else.

Log(a(n))/n^2 approaches a constant (approximately 0.24) as n approaches infinity.

This limit is equal to log(phi)/2 = 0.24060591252980172374887945671218421156759..., where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Apr 10 2016

Crossrefs

Cf. A001610, A003266, A000032.

Programs

Mathematica

Table[Det[Array[1+(LucasL[#1]-1)*KroneckerDelta[#1,#2]&,{n,n}]],{n,30}] (* or *)
Table[Product[Fibonacci[k]+Fibonacci[k+2]-1,{k,1,n-1}],{n,30}] (* or *)
RecurrenceTable[{a[n+1]==(Fibonacci[n]+Fibonacci[n+2]-1) a[n], a[1] == 1}, a, {n, 30}]

Formula

prod(F(k)+F(k+2)-1, k=1..n-1), where F(k) is the k-th Fibonacci number.

a(n) ~ c * phi^(n*(n+1)/2), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio and c = 0.22805409619361969822736866017363184926893729185052240813641180656087... . - Vaclav Kotesovec, Apr 10 2016