%I #19 Jan 01 2024 08:33:59

%S 1,3,18,90,468,2412,12456,64296,331920,1713456,8845344,45662112,

%T 235720512,1216854720,6281741952,32428096128,167402836224,

%U 864179921664,4461136704000,23029626345984,118885325607936,613719060507648

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

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

%C 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

%H G. C. Greubel, <a href="/A147518/b147518.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,6).

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

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

%F a(n) = ((10+sqrt(10))/20)*(2+sqrt(10))^n + ((10-sqrt(10))/20)*(2-sqrt(10))^n. - _Richard Choulet_, Nov 20 2008

%p 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

%t 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 *)

%o (PARI) my(x='x+O('x^30)); Vec((1-x)/(1-4*x-6*x^2)) \\ _G. C. Greubel_, Jan 09 2020

%o (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

%o (Sage)

%o def A147518_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( (1-x)/(1-4*x-6*x^2) ).list()

%o A147518_list(30) # _G. C. Greubel_, Jan 09 2020

%o (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

%Y Cf. A026150, A122117.

%K nonn

%O 0,2

%A _Philippe Deléham_, Nov 06 2008