A081076 a(n) = Lucas(4n) + 3, or 5*Fibonacci(2n-1)*Fibonacci(2n+1).

5, 10, 50, 325, 2210, 15130, 103685, 710650, 4870850, 33385285, 228826130, 1568397610, 10749957125, 73681302250, 505019158610, 3461452808005, 23725150497410, 162614600673850, 1114577054219525, 7639424778862810

Offset: 0

Author

R. K. Guy, Mar 04 2003

Keywords

• nonn
• easy

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Crossrefs

Cf. A000032 (Lucas numbers), A000045 (Fibonacci numbers), A056854.

Programs

GAP

List([0..30], n-> Lucas(1, -1, 4*n)[2] +3 ); # G. C. Greubel, May 26 2020

Magma

[Lucas(4*n) +3: n in [0..30]]; // G. C. Greubel, May 26 2020

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 30 do printf(`%d,`,luc(4*n)+3) od: # James Sellers, Mar 05 2003

Mathematica

Table[LucasL[4 n] + 3, {n,0,30}] (* Wesley Ivan Hurt, Nov 20 2014 *)

PARI

Vec(-5*(2*x^2-6*x+1)/((x-1)*(x^2-7*x+1)) + O(x^30)) \\ Michel Marcus, Dec 23 2014

Sage

[lucas_number2(4*n,1,-1) + 3 for n in (0..30)] # G. C. Greubel, May 26 2020

Formula

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

G.f.: 5*(1 -6*x +2*x^2)/((1-x)*(1-7*x+x^2)). - Colin Barker, Jun 22 2012

Sum_{n>=0} 1/a(n) = phi/5, where phi = A001622 = (1 + sqrt(5))/2. - Diego Rattaggi, Apr 22 2020

From G. C. Greubel, May 26 2020: (Start)

a(n) = 5*A064170(n+1).

a(n) = Lucas(n)^4 - 4*(-1)^n*Lucas(n)^2 + 5.

E.g.f.: 3*exp(x) + 2*exp(7*x/2)*cosh(3*sqrt(5)*x/2). (End)

Extensions

More terms from James Sellers, Mar 05 2003