A147518 Expansion of (1-x)/(1-4*x-6*x^2).

1, 3, 18, 90, 468, 2412, 12456, 64296, 331920, 1713456, 8845344, 45662112, 235720512, 1216854720, 6281741952, 32428096128, 167402836224, 864179921664, 4461136704000, 23029626345984, 118885325607936, 613719060507648

Offset: 0

Author

Philippe Deléham, Nov 06 2008

Keywords

• nonn

Comments

Binomial transform of [1,2,13,44,205,...] = A002534(n+1).

a(n) is the number of compositions of n when there are 3 types of 1 and 9 types of other natural numbers. - Milan Janjic, Aug 13 2010

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,6).

Crossrefs

Cf. A026150, A122117.

Programs

GAP

a:=[1,3];; for n in [3..30] do a[n]:=4*a[n-1]+6*a[n-2]; od; a; # G. C. Greubel, Jan 09 2020

Magma

I:=[1,3]; [n le 2 select I[n] else 4*Self(n-1) + 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 09 2020

Maple

seq(coeff(series((1-x)/(1-4*x-6*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 09 2020

Mathematica

CoefficientList[Series[(1-x)/(1-4x-6x^2),{x,0,30}],x] (* or *) LinearRecurrence[{4,6},{1,3},30] (* Harvey P. Dale, Aug 22 2016 *)

PARI

my(x='x+O('x^30)); Vec((1-x)/(1-4*x-6*x^2)) \\ G. C. Greubel, Jan 09 2020

Sage

def A147518_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-4*x-6*x^2) ).list()
A147518_list(30) # G. C. Greubel, Jan 09 2020

Formula

a(n) = 4*a(n-1) + 6*a(n-2) with a(0)=1, a(1)=3.

a(n) = Sum_{k=0..n} A122016(n,k)*3^k.

a(n) = ((10+sqrt(10))/20)*(2+sqrt(10))^n + ((10-sqrt(10))/20)*(2-sqrt(10))^n. - Richard Choulet, Nov 20 2008