OFFSET
0,2
COMMENTS
a(n+1) is the Hankel transform of A051960 aerated.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,3,-1).
FORMULA
From Peter Bala, May 01 2012: (Start)
a(n) = (-1)^n + 3*Sum_{k = 1..n} (-1)^(n-k)*6^(k-1)*binomial(n+k,2*k).
a(n) = (-1)^n*R(n,-3), where R(n,x) is the n-th row polynomial of A211955.
a(n) = (-1)^n*1/u*T(n,u)*T(n+1,u) with u = sqrt(-1/2) and T(n,x) denotes the Chebyshev polynomial of the first kind Cf. A182432.
Recurrence: a(n) = 4*a(n-1) -a(n-2) +3*(-1)^n, with a(0) = 1 and a(1) = 2; a(n)*a(n-2) = a(n-1)*(a(n-1)+3*(-1)^n).
Sum_{k>=0} (-1)^k/a(k) = 1/sqrt(3). (End)
From Colin Barker, Jul 29 2013: (Start)
a(n) = 3*a(n-1) + 3*a(n-2) - a(n-3).
G.f.: (1-x+x^2)/((1+x)*(1-4*x+x^2)). (End)
MAPLE
seq(coeff(series((1-x+x^2)/((1+x)*(1-4*x+x^2)), x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 09 2020
MATHEMATICA
LinearRecurrence[{3, 3, -1}, {1, 2, 10}, 30] (* G. C. Greubel, Jan 09 2020 *)
PROG
(PARI) my(x='x+O('x^30)); Vec((1-x+x^2)/((1+x)*(1-4*x+x^2))) \\ G. C. Greubel, Jan 09 2020
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x+x^2)/((1+x)*(1-4*x+x^2)) )); // G. C. Greubel, Jan 09 2020
(Sage)
def A146983_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1-x+x^2)/((1+x)*(1-4*x+x^2)) ).list()
A146983_list(30) # G. C. Greubel, Jan 09 2020
(GAP) a:=[1, 2, 10];; for n in [4..30] do a[n]:=3*a[n-1]+3*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Jan 09 2020
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Nov 04 2008
EXTENSIONS
More terms from Colin Barker, Jul 29 2013
STATUS
approved