A146983 a(n) = A002531(n)*A002531(n+1).

1, 2, 10, 35, 133, 494, 1846, 6887, 25705, 95930, 358018, 1336139, 4986541, 18610022, 69453550, 259204175, 967363153, 3610248434, 13473630586, 50284273907, 187663465045, 700369586270, 2613814880038, 9754889933879, 36405744855481, 135868089488042

Offset: 0

Author

Paul Barry, Nov 04 2008

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A001654, A182432, A211955.

Programs

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

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x+x^2)/((1+x)*(1-4*x+x^2)) )); // G. C. Greubel, Jan 09 2020

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 *)

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

Sage

def A146983_list(prec):
    P. = 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

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)

Extensions

More terms from Colin Barker, Jul 29 2013