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%I #16 Mar 25 2021 21:19:14
%S 1,2,8,30,114,432,1638,6210,23544,89262,338418,1283040,4864374,
%T 18442242,69919848,265086270,1005018354,3810313872,14445996678,
%U 54768931650,207644784984,787241149902,2984657804658,11315696863680,42901064005014
%N a(n) = 3*a(n-1) + 3*a(n-2), n>2, a(0)=1, a(1)=2, a(2)=8.
%C From _Johannes W. Meijer_, Aug 14 2010: (Start)
%C A berserker sequence, see A180140 and A180147. For the central square 16 A[5] vectors with decimal values between 3 and 384 lead to this sequence. These vectors lead for the corner squares to A123620 and for the side squares to A180142.
%C This sequence belongs to a family of sequences with GF(x)=(1-(2*k-1)*x-k*x^2)/(1-3*x+(k-4)*x^2). Berserker sequences that are members of this family are A000007 (k=2), A155116 (k=1; this sequence), A000302 (k=0), 6*A179606 (k=-1; with leading 1 added) and 2*A180141 (k=-2; n>=1 and a(0)=1). Some other members of this family are (-2)*A003688 (k=3; with leading 1 added), (-4)*A003946 (k=4; with leading 1 added), (-6)*A002878 (k=5; with leading 1 added) and (-8)*A033484 (k=6; with leading 1 added).
%C Inverse binomial transform of A101368 (without the first leading 1).
%C (End)
%H G. C. Greubel, <a href="/A155116/b155116.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 (3,3).
%F G.f.: (1-x-x^2)/(1-3*x-3*x^2).
%F a(n) = 2*A125145(n-1), n>=1 .
%F a(n) = ( (2+4*A)*A^(-n-1) + (2+4*B)*B^(-n-1) )/21 with A=(-3+sqrt(21))/6 and B=(-3-sqrt(21))/6 for n>=1 with a(0)=1. [corrected by _Johannes W. Meijer_, Aug 12 2010]
%F Contribution from _Johannes W. Meijer_, Aug 14 2010: (Start)
%F a(n) = A123620(n)/2 for n>=1.
%F (End)
%F a(n) = (1/3)*[n=0] - 2*(sqrt(3)*i)^(n-2)*ChebyshevU(n, -sqrt(3)*i/2). - _G. C. Greubel_, Mar 25 2021
%t With[{m=3}, LinearRecurrence[{m, m}, {1, m-1, m^2-1}, 30]] (* _G. C. Greubel_, Mar 25 2021 *)
%o (PARI) Vec((1-x-x^2)/(1-3*x-3*x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Jan 12 2012
%o (Magma) m:=3; [1] cat [n le 2 select (m-1)*(m*n-(m-1)) else m*(Self(n-1) + Self(n-2)): n in [1..30]]; // _G. C. Greubel_, Mar 25 2021
%o (Sage) m=3; [1]+[-(m-1)*(sqrt(m)*i)^(n-2)*chebyshev_U(n, -sqrt(m)*i/2) for n in (1..30)] # _G. C. Greubel_, Mar 25 2021
%Y Sequences of the form a(n) = m*(a(n-1) + a(n-2)) with a(0)=1, a(1) = m-1, a(2) = m^2 -1: A155020 (m=2), this sequence (m=3), A155117 (m=4), A155119 (m=5), A155127 (m=6), A155130 (m=7), A155132 (m=8), A155144 (m=9), A155157 (m=10).
%K nonn,easy
%O 0,2
%A _Philippe Deléham_, Jan 20 2009
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