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%I #9 Apr 07 2020 21:37:10
%S 2,4,5,7,9,12,17,23,33,48,70,103,152,228,343,515,779,1180,1787,2715,
%T 4124,6264,9526,14483,22025,33504,50957,77519,117929,179396,272930,
%U 415215,631680,961032,1462067,2224347,3384083,5148432,7832727,11916547,18129540
%N a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - 4*a(n-4) + 4*a(n-5) - 4*a(n-6) + 4*a(n-7) - 4*a(n-8) + 4*a(n-9) - 3*a(n-10) + 2*a(n-11) - 3*a(n-12) + 2*a(n-13) for n >= 16, with initial values as shown.
%C Conjecture: a(n) is the number of letters (0's and 1's) in the n-th iteration of the mapping 00->0101, 10->001, starting with 00; see A288508.
%H Clark Kimberling, <a href="/A288511/b288511.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (2, -1, 2, -4, 4, -4, 4, -4, 4, -3, 2, -3, 2).
%F G.f.: x*(2 - x^2 - 3*x^3 - x^5 - x^7 - x^9 - x^10 - 3*x^11 + x^12 + 2*x^13 + x^14- 2*x^15) / ((1 - x)^2*(1 - x^2 - 2*x^3)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)). - _Colin Barker_, Jun 12 2017
%F a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - 4*a(n-4) + 4*a(n-5) - 4*a(n-6) + 4*a(n-7) - 4*a(n-8) + 4*a(n-9) - 3*a(n-10) + 2*a(n-11) - 3*a(n-12) + 2*a(n-13) for n >= 16, where a(0)=2, a(1)=4, a(2)=5, a(3)=7, a(4)=9, a(5)=12, a(6)=17, a(7)=23, a(8)=33, a(9)=48, a(10)=70, a(11)=103, a(12)=152, a(13)=228, a(14)=343, a(15)=515.
%t Join[{2, 4, 5}, LinearRecurrence[{2, -1, 2, -4, 4, -4, 4, -4, 4, -3, 2, -3, 2}, {7, 9, 12, 17, 23, 33, 48, 70, 103, 152, 228, 343, 515}, 40]]
%o (PARI) Vec(x*(2 - x^2 - 3*x^3 - x^5 - x^7 - x^9 - x^10 - 3*x^11 + x^12 + 2*x^13 + x^14- 2*x^15) / ((1 - x)^2*(1 - x^2 - 2*x^3)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)) + O(x^60)) \\ _Colin Barker_, Jun 12 2017
%Y Cf. A288508.
%K nonn,easy
%O 0,1
%A _Clark Kimberling_, Jun 12 2017
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