A099133 4^(n-1)*Fibonacci(n).

0, 1, 4, 32, 192, 1280, 8192, 53248, 344064, 2228224, 14417920, 93323264, 603979776, 3909091328, 25300041728, 163745628160, 1059783180288, 6859062771712, 44392781971456, 287316132233216, 1859549040476160, 12035254277636096, 77893801758162944

Offset: 0

Author

Paul Barry, Sep 29 2004

Keywords

• nonn
• easy

Comments

Binomial transform of A099134.

Second binomial transform of x/(1-20x^2), or (0,1,0,20,0,400,0,8000,....).

In general k^(n-1)*Fibonacci(n) has g.f. x/(1-kx-k^2x^2).

The ratio a(n+1)/a(n) converges to 4 times the golden ratio as n approaches infinity. In general, the ratio a(n+1)/a(n) of the sequence which is the solution to the linear recurrence relation a(n) = m*a(n-1)+m^2*a(n-2) with a(0)=0 and a(1) = 1 converges to m times the golden ratio as n approaches infinity where m is a positive integer. - Felix P. Muga II, Mar 10 2014

Examples

			G.f. = x + 4*x^2 + 32*x^3 + 192*x^4 + 1280*x^5 + 8192*x^6 + 53248*x^7 + ...
		

References

• F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivré Formula, March 2014; Preprint on ResearchGate.

Links

• Index entries for linear recurrences with constant coefficients, signature (4,16).

Crossrefs

Cf. A000045, A099012, A085449. Fourth row of A234357.

Programs

Mathematica

Join[{a=0,b=1},Table[c=4*b+16*a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Mar 29 2011*)
Table[4^(n-1) Fibonacci[n],{n,0,20}] (* Harvey P. Dale, Aug 22 2012 *)
LinearRecurrence[{4,16},{0,1},30] (* Harvey P. Dale, Aug 22 2012 *)

PARI

a(n) = 4^(n-1)*fibonacci(n); \\ Michel Marcus, Jan 10 2014

Formula

G.f.: x/(1-4*x-16*x^2).

a(n) = 4*a(n-1) + 16*a(n-2).

a(n) = (2+2*sqrt(5))^n/(4*sqrt(5))-(2-sqrt(5))^n/(4*sqrt(5)).

a(-n) = -(-1)^n * a(n) / 16^n for all n in Z. - Michael Somos, Mar 18 2014