OFFSET
0,3
COMMENTS
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
J. Pan, Multiple Binomial Transforms and Families of Integer Sequences , J. Int. Seq. 13 (2010), 10.4.2, F^(4).
Index entries for linear recurrences with constant coefficients, signature (9,-19).
FORMULA
a(n) = 9*a(n-1) - 19*a(n-1), a(0)=0, a(1)=1.
a(n) = ((sqrt(5)/2 + 9/2)^n - (9/2 - sqrt(5)/2)^n)/sqrt(5).
G.f.: x/(1 - 9*x + 19*x^2).
E.g.f.: 2*exp(9*x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - Ilya Gutkovskiy, Aug 11 2017
MAPLE
seq(coeff(series(x/(1-9*x+19*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Aug 13 2019
MATHEMATICA
Join[{a=0, b=1}, Table[c=9*b-19*a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2011 *)
LinearRecurrence[{9, -19}, {0, 1}, 30] (* Harvey P. Dale, Dec 03 2011 *)
CoefficientList[Series[x/(1 -9x +19x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 09 2013 *)
PROG
(Sage) [lucas_number1(n, 9, 19) for n in range(0, 21)] # Zerinvary Lajos, Apr 23 2009
(Magma) [n le 2 select (n-1) else 9*Self(n-1)-19*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 09 2013
(PARI) my(x='x+O('x^30)); Vec(x/(1 - 9*x + 19*x^2)) \\ G. C. Greubel, Aug 13 2019
(GAP) a:=[0, 1];; for n in [3..30] do a[n]:=9*a[n-1]-19*a[n-2]; od; a; # G. C. Greubel, Aug 13 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Mar 22 2003
EXTENSIONS
Corrected by Philippe Deléham, Dec 16 2009
STATUS
approved