OFFSET
1,3
COMMENTS
Based on the observation that F_{n+1} = Sum_{k} binomial (n-k,k). In both cases the sum is extended to 0<=2k<=n.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1,1).
FORMULA
G.f.: x*(1-x-x^2)/((1-x)*(1-x-2*x^2+x^4)).
a(1)=1, a(2)=1, a(3)=2, a(4)=3, a(5)=5, a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Harvey P. Dale, Jun 23 2011
MAPLE
seq(coeff(series(x*(1-x-x^2)/((1-x)*(1-x-2*x^2+x^4)), x, n+1), x, n), n = 1..40); # G. C. Greubel, Aug 26 2019
MATHEMATICA
Rest@CoefficientList[ Series[x(1-x-x^2)/((1-x)(1-x-2x^2+x^4)), {x, 0, 40}], x] (* or *) LinearRecurrence[{2, 1, -2, -1, 1}, {1, 1, 2, 3, 5}, 40] (* Harvey P. Dale, Jun 23 2011 *)
PROG
(PARI) my(x='x+O('x^40)); Vec(x*(1-x-x^2)/((1-x)*(1-x-2*x^2+x^4))) \\ G. C. Greubel, Aug 26 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x*(1-x-x^2)/((1-x)*(1-x-2*x^2+x^4)) )); // G. C. Greubel, Aug 26 2019
(Sage)
def A072176_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x*(1-x-x^2)/((1-x)*(1-x-2*x^2+x^4)) ).list()
a=A072176_list(40); a[1:] # G. C. Greubel, Aug 26 2019
(GAP) a:=[1, 1, 2, 3, 5];; for n in [6..40] do a[n]:=2*a[n-1]+a[n-2] -2*a[n-3]-a[n-4]+a[n-5]; od; a; # G. C. Greubel, Aug 26 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michele Dondi (bik.mido(AT)tiscalinet.it), Jun 30 2002
STATUS
approved