%I #12 Apr 07 2020 21:47:10

%S 2,4,8,18,40,92,210,484,1112,2562,5896,13580,31266,72004,165800,

%T 381810,879208,2024636,4662258,10736164,24722936,56931426,131100232,

%U 301894508,695195202,1600878724,3686464328,8489100498,19548493480,45015794972,103661275410

%N a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3), where a(0) = 2, a(1) = 4, a(2) = 8.

%C Conjecture: a(n) is the number of letters (0's and 1's) in the n-th iteration of the mapping 00->0010, 1->110, starting with 00; see A288306.

%H Clark Kimberling, <a href="/A288309/b288309.txt">Table of n, a(n) for n = 0..2000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2, 2, -3).

%F a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3), where a(0) = 2, a(1) = 4, a(2) = 8.

%F G.f.: -((2*(-1 + 2*x^2))/(1 - 2*x - 2*x^2 + 3*x^3)).

%F a(n) = (2^(1-n)*(13*2^n + (13-4*sqrt(13))*(1-sqrt(13))^n + (1+sqrt(13))^n*(13+4*sqrt(13)))) / 39. - _Colin Barker_, Jun 09 2017

%t LinearRecurrence[{2, 2, -3}, {2, 4, 8}, 40]

%o (PARI) Vec(2*(1 - 2*x^2) / ((1 - x)*(1 - x - 3*x^2)) + O(x^30)) \\ _Colin Barker_, Jun 09 2017

%Y Cf. A288306.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Jun 09 2017