OFFSET
0,3
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
REFERENCES
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivré Formula, March 2014; Preprint on ResearchGate.
LINKS
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
EXAMPLE
G.f. = x + 4*x^2 + 32*x^3 + 192*x^4 + 1280*x^5 + 8192*x^6 + 53248*x^7 + ...
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 *)
PROG
(PARI) a(n) = 4^(n-1)*fibonacci(n); \\ Michel Marcus, Jan 10 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Sep 29 2004
STATUS
approved