0,3

Binomial transform is A099133. Binomial transform of x/(1-20*x^2), or (0,1,0,20,0,400,0,8000,....). The inverse binomial transform of k^(n-1)*Fibonacci(n) has g.f. x/(1-(k-2)*x-(k^2+k-1)*x^2).

4*a(n) = (-1)^(n+1)*b(n;4) = 3^n*b(n;4/3), where b(n;d), n=0,1,..., d \in C, denote one of the delta-Fibonacci numbers defined in comments to A014445 (see also Wituła-Słota's paper). Our first identity is equivalent to the second formula given below. We note that the sequence (4/3)^n*Fibonacci(n) is the binomial transform of the sequence 3^(-n)*b(n;4). - Roman Witula, Jul 24 2012

Roman Wituła and Damian Słota, delta-Fibonacci numbers, Appl. Anal. Discr. Math 3 (2009), 310-329, MR2555042.

Index entries for linear recurrences with constant coefficients, signature (2,19).

a(n) = 2*a(n-1) + 19*a(n-2).

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*4^(k-1)*Fib(k).

a(n) = Sum_{k=0..n} binomial(n, 2*k+1)*20^k.

From Amiram Eldar, Jan 21 2026: (Start)

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

E.g.f.: exp(x) * sinh(2*sqrt(5)*x) / (2*sqrt(5)). (End)

Join[{a=0, b=1}, Table[c=2*b+19*a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)

CoefficientList[Series[x/(1-2x-19x^2), {x, 0, 30}], x] (* or *) LinearRecurrence[ {2, 19}, {0, 1}, 30] (* Harvey P. Dale, Dec 25 2019 *)

Cf. A000045, A014445, A015447, A099133.

Sequence in context: A097232 A339474 A222564 * A069152 A131464 A245331

Adjacent sequences: A099131 A099132 A099133 * A099135 A099136 A099137

easy,nonn

Paul Barry, Sep 29 2004

approved