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

%I A147518 #19 Jan 01 2024 08:33:59
%S A147518 1,3,18,90,468,2412,12456,64296,331920,1713456,8845344,45662112,
%T A147518 235720512,1216854720,6281741952,32428096128,167402836224,
%U A147518 864179921664,4461136704000,23029626345984,118885325607936,613719060507648
%N A147518 Expansion of (1-x)/(1-4*x-6*x^2).
%C A147518 Binomial transform of [1,2,13,44,205,...] = A002534(n+1).
%C A147518 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 A147518 G. C. Greubel, <a href="/A147518/b147518.txt">Table of n, a(n) for n = 0..1000</a>
%H A147518 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,6).
%F A147518 a(n) = 4*a(n-1) + 6*a(n-2) with a(0)=1, a(1)=3.
%F A147518 a(n) = Sum_{k=0..n} A122016(n,k)*3^k.
%F A147518 a(n) = ((10+sqrt(10))/20)*(2+sqrt(10))^n + ((10-sqrt(10))/20)*(2-sqrt(10))^n. - _Richard Choulet_, Nov 20 2008
%p A147518 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 A147518 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 A147518 (PARI) my(x='x+O('x^30)); Vec((1-x)/(1-4*x-6*x^2)) \\ _G. C. Greubel_, Jan 09 2020
%o A147518 (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 A147518 (Sage)
%o A147518 def A147518_list(prec):
%o A147518     P.<x> = PowerSeriesRing(ZZ, prec)
%o A147518     return P( (1-x)/(1-4*x-6*x^2) ).list()
%o A147518 A147518_list(30) # _G. C. Greubel_, Jan 09 2020
%o A147518 (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 A147518 Cf. A026150, A122117.
%K A147518 nonn
%O A147518 0,2
%A A147518 _Philippe Deléham_, Nov 06 2008