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A099842 Expansion of (1-x)/(1 + 6*x - 3*x^2).

%I A099842 #28 Aug 31 2024 13:54:26
%S A099842 1,-7,45,-291,1881,-12159,78597,-508059,3284145,-21229047,137226717,
%T A099842 -887047443,5733964809,-37064931183,239591481525,-1548743682699,
%U A099842 10011236540769,-64713650292711,418315611378573,-2704034619149571,17479154549033145,-112987031151647583
%N A099842 Expansion of (1-x)/(1 + 6*x - 3*x^2).
%C A099842 A transformation of x/(1-2*x-2*x^2).
%C A099842 The g.f. is the transform of the g.f. of A002605 under the mapping G(x) -> (-1/(1+x))*G((x-1)/(x+1)). In general this mapping transforms x/(1-k*x-k*x^2) into (1-x)/(1+2*(k+1)*x-(2*k-1)*x^2).
%C A099842 For n >= 1, |a(n)| equals the numbers of words of length n-1 on alphabet {0,1,...,6} containing no subwords 00, 11, 22, 33. - _Milan Janjic_, Jan 31 2015
%H A099842 G. C. Greubel, <a href="/A099842/b099842.txt">Table of n, a(n) for n = 0..1000</a>
%H A099842 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-6,3).
%F A099842 G.f.: (1-x)/(1+6*x-3*x^2).
%F A099842 a(n) = (1/2 - sqrt(3)/3)*(-3 + 2*sqrt(3))^n + (1/2 + sqrt(3)/3)*(-3 - 2*sqrt(3))^n.
%F A099842 a(n) = (-1)^n*Sum_{k=0..n} binomial(n, k)(-1)^(n-k)*A002605(2k+2)/2.
%F A099842 a(n) = (-1)^n*(A090018(n) + A090018(n-1)). - _R. J. Mathar_, Apr 07 2022
%t A099842 LinearRecurrence[{-6,3}, {1,-7}, 31] (* _G. C. Greubel_, Oct 10 2022 *)
%t A099842 CoefficientList[Series[(1-x)/(1+6x-3x^2),{x,0,40}],x] (* _Harvey P. Dale_, Aug 31 2024 *)
%o A099842 (Magma) [n le 2 select (-7)^(n-1) else -6*Self(n-1) +3*Self(n-2): n in [1..31]]; // _G. C. Greubel_, Oct 10 2022
%o A099842 (SageMath)
%o A099842 A099842 = BinaryRecurrenceSequence(-6,3,1,-7)
%o A099842 [A099842(n) for n in range(31)] # _G. C. Greubel_, Oct 10 2022
%Y A099842 Cf. A002605, A090018.
%K A099842 easy,sign
%O A099842 0,2
%A A099842 _Paul Barry_, Oct 27 2004