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%I #23 Oct 03 2021 08:43:54
%S 2,4,8,16,34,72,152,324,690,1468,3128,6664,14194,30240,64424,137244,
%T 292386,622900,1327016,2827072,6022786,12830904,27334904,58234164,
%U 124061778,264300652,563064920,1199550904,2555517778,5444263440,11598433928,24709250700
%N a(n) = 2*a(n-1) + 2*a(n-3) - 3*a(n-4), where a(0) = 2, a(1) = 4, a(2) = 8, a(3) = 16.
%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->100, starting with 00; see A288257. [Mapping corrected by _Michel Dekking_, Mar 05 2020.]
%C Second Conjecture: a(n) is the number of letters (0's and 1's) in the n-th iteration of the mapping 00->0101, 1->`010, starting with 00; see A288466. - _Michel Dekking_, Mar 05 2020
%H Clark Kimberling, <a href="/A288260/b288260.txt">Table of n, a(n) for n = 0..2000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,2,-3).
%F a(n) = 2*a(n-1) + 2*a(n-3) - 3*a(n-4), where a(0) = 2, a(1) = 4, a(2) = 8, a(3) = 16.
%F G.f.: -2*(-1+2*x^3)/(x-1)/(3*x^3+x^2+x-1) .
%t LinearRecurrence[{2, 0, 2, -3}, {2, 4, 8, 16}, 40]
%Y Cf. A288257, A288466.
%K nonn,easy
%O 0,1
%A _Clark Kimberling_, Jun 09 2017
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