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%I #10 Jan 03 2024 08:47:30
%S 1,3,15,72,348,1680,8112,39168,189120,913152,4409088,21288960,
%T 102792192,496324608,2396467200,11571167232,55870537728,269766819840,
%U 1302549430272,6289265000448,30367257722880,146626090893312
%N a(n) = 4*a(n-1) + 4*a(n-2), n>2, a(0)=1, a(1)=3, a(2)=15.
%H G. C. Greubel, <a href="/A155117/b155117.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: (1-x-x^2)/(1-4*x-4*x^2) . a(n)=3*A086347(n), n>=1 .
%F From _G. C. Greubel_, Mar 25 2021: (Start)
%F a(n) = (1/4)*[n=0] - 3*(2*i)^(n-2)*ChebyshevU(n, -i).
%F a(n) = (1/4)*[n=0] + 3*2^(n-2)*P_{n+1}, where P_{n} = A000129(n) (Pell numbers). (End)
%p 1,seq(simplify(-3*(2*I)^(n-2)*ChebyshevU(n, -I)), n = 1..30); # _G. C. Greubel_, Mar 25 2021
%t With[{m=4}, LinearRecurrence[{m, m}, {1, m-1, m^2-1}, 30]] (* _G. C. Greubel_, Mar 25 2021 *)
%o (Magma) m:=4; [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=4; [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), A155116 (m=3), this sequence (m=4), A155119 (m=5), A155127 (m=6), A155130 (m=7), A155132 (m=8), A155144 (m=9), A155157 (m=10).
%Y Cf. A000129.
%K nonn
%O 0,2
%A _Philippe Deléham_, Jan 20 2009
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