0,3

Binomial transform of A099134.

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

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

Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivré Formula, ResearchGate preprint, March 2014.

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

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

E.g.f.: exp(2*x) * sinh(2*sqrt(5)*x) / (2*sqrt(5)). - Amiram Eldar, Jan 15 2026

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

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

Fourth row of A234357.

Cf. A000045, A099012, A085449.

Sequence in context: A271462 A270178 A018215 * A208710 A043018 A270015

Adjacent sequences: A099130 A099131 A099132 * A099134 A099135 A099136

nonn,easy

Paul Barry, Sep 29 2004

approved