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A163063 Lucas(3n+2) = Fibonacci(3n+1) + Fibonacci(3n+3).

%I A163063 #31 Sep 08 2022 08:45:46
%S A163063 3,11,47,199,843,3571,15127,64079,271443,1149851,4870847,20633239,
%T A163063 87403803,370248451,1568397607,6643838879,28143753123,119218851371,
%U A163063 505019158607,2139295485799,9062201101803,38388099893011
%N A163063 Lucas(3n+2) = Fibonacci(3n+1) + Fibonacci(3n+3).
%C A163063 Binomial transform of A163062. Second binomial transform of A163114. Inverse binomial transform of A098648 without initial 1.
%H A163063 Nathaniel Johnston, <a href="/A163063/b163063.txt">Table of n, a(n) for n = 0..400</a>
%H A163063 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,1).
%F A163063 a(n) = 4*a(n-1)+a(n-2) for n > 1; a(0) = 3, a(1) = 11.
%F A163063 G.f.: (3-x)/(1-4*x-x^2).
%F A163063 a(n) = A033887(n) + A014445(n+1).
%F A163063 a(n) = ((3+sqrt(5))*(2+sqrt(5))^n+(3-sqrt(5))*(2-sqrt(5))^n)/2.
%F A163063 a(n) = A000032(3*n+2), n>=0, (Lucas trisection). - _Wolfdieter Lang_, Mar 09 2011.
%F A163063 a(n) = 5*F(n)*F(n+1)*L(n+1) + L(n+2)*(-1)^n with F(n)=A000045(n) and L(n)=A000032(n). - _J. M. Bergot_, Dec 10 2015
%p A163063 with(combinat):A163063:=proc(n)return fibonacci(3*n+1) + fibonacci(3*n+3): end:seq(A163063(n), n=0..21); # _Nathaniel Johnston_, Apr 18 2011
%t A163063 Table[Fibonacci[3n + 1] + Fibonacci[3n + 3], {n, 0, 21}] (* _Alonso del Arte_, Nov 29 2010 *)
%t A163063 LinearRecurrence[{4,1},{3,11},30] (* _Harvey P. Dale_, Apr 14 2021 *)
%o A163063 (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-5); S:=[ ((3+r)*(2+r)^n+(3-r)*(2-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jul 21 2009
%o A163063 (Magma) [Lucas(3*n+2): n in [0..30]]; // _Vincenzo Librandi_, Apr 18 2011
%o A163063 (PARI) Vec((3-x)/(1-4*x-x^2) + O(x^100)) \\ _Altug Alkan_, Dec 10 2015
%Y A163063 Cf. A000032, A000045, A163062, A163114, A098648, A001077 (L(3*n)/L(2)), A048876 (L(3*n+1)).
%K A163063 nonn,easy
%O A163063 0,1
%A A163063 Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009
%E A163063 Edited and extended beyond a(5) by _Klaus Brockhaus_, Jul 21 2009