A081067 a(n) = Lucas(4n+2)+2, or 5*Fibonacci(2n+1)^2.

5, 20, 125, 845, 5780, 39605, 271445, 1860500, 12752045, 87403805, 599074580, 4106118245, 28143753125, 192900153620, 1322157322205, 9062201101805, 62113250390420, 425730551631125, 2918000611027445, 20000273725560980

Offset: 0

Author

R. K. Guy, Mar 04 2003

Keywords

• nonn
• easy

Comments

a(n) is the square of limit of (G(j+2n-1) + G(j-2n+1))/G(j) as j -> infinity, where G(n) is any sequence of the form G(n+1) = G(n) + G(n-1), with any starting values, including non-integer values. G(n) includes Lucas and Fibonacci. Compare with A005248 for even number offsets from j in any such G(n). - Richard R. Forberg, Nov 16 2014

a(n) = (t(i+6n+3) + t(i))/(t(i+4n+2) + t(i+2n+1)) + 3, where (t) is any sequence of the form t(n+2) = 2t(n+1) + 2t(n) - t(n-1) or of the form t(n+1) = 3t(n) - t(n-1) without regard to initial values as long as t(i+4n+2) + t(i+2n+1) != 0. - Klaus Purath, Jun 23 2024

References

• Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.

Links

• Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Crossrefs

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers).

Programs

Maple

luc := proc(n) option remember: if n=0 then RETURN(2) fi: if n=1 then RETURN(1) fi: luc(n-1)+luc(n-2): end: for n from 0 to 40 do printf(`%d,`,luc(4*n+2)+2) od: # James Sellers, Mar 05 2003

Mathematica

Table[LucasL[4n+2]+2,{n,0,20}] (* or *)
Table[5Fibonacci[2n+1]^2,{n,0,30}] (* Harvey P. Dale, Apr 18 2011 *)

PARI

a(n)=5*fibonacci(2*n+1)^2 \\ Charles R Greathouse IV, Nov 17 2014

Formula

a(n) = 8a(n-1) - 8a(n-2) + a(n-3).

G.f.: -5*(x^2-4*x+1)/((x-1)*(x^2-7*x+1)). - Colin Barker, Jun 25 2012

a(n) ~ phi^(4n+2). - Charles R Greathouse IV, Nov 17 2014

a(n) = 5*A081068(n). - R. J. Mathar, Feb 13 2016