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%I #15 Jul 02 2023 18:16:03
%S 3,3,7,12,23,43,82,157,303,588,1147,2247,4418,8713,17227,34132,67743,
%T 134643,267922,533637,1063703,2121628,4233907,8452687,16880898,
%U 33722193,67380307,134656932,269146103,538020763,1075602322,2150493997
%N a(n) = L(n) + 2^n where L(n) = A000032(n) (the Lucas numbers).
%C Lim_{n->infinity} a(n)/a(n-1) = 2.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, -1, -2).
%F G.f.: (3 - 6*x + 2*x^2) / (1 - 3*x + x^2 + 2*x^3)
%F a(n) = p^n + q^n + r^n, where p = (1+sqrt(5))/2, q = (1-sqrt(5))/2, and r = 2*p^n + q^n = L(n) = A000032(n), so a(n) = L(n) + 2^n
%F a(0)=3, a(1)=3, a(2)=7 and a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3) for n >= 3.
%e a(6) = 82 = L(6) + 2^6 = 18 + 64.
%e a(7) = 157 = 3*82 - 43 - 2*23 = 246 - 43 - 46.
%o (Magma) [2^n+Lucas(n): n in [0..50]]; // _Vincenzo Librandi_, Apr 14 2011
%Y Cf. A000032.
%K easy,nonn
%O 0,1
%A _Miklos Kristof_, Nov 25 2003
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