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A289131 a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 2*a(n-4) + 2*a(n-5) for n >= 6, a(0) = 2, a(1) = 4, a(2) = 7, a(3) = 11, a(4) = 18, a(5) = 28. 2

%I #19 Apr 07 2020 21:28:35

%S 2,4,7,11,18,28,43,65,96,142,205,299,426,616,871,1253,1764,2530,3553,

%T 5087,7134,10204,14299,20441,28632,40918,57301,81875,114642,163792,

%U 229327,327629,458700,655306,917449,1310663,1834950,2621380,3669955,5242817,7339968

%N a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 2*a(n-4) + 2*a(n-5) for n >= 6, a(0) = 2, a(1) = 4, a(2) = 7, a(3) = 11, a(4) = 18, a(5) = 28.

%C Conjecture: a(n) is the number of letters (0's and 1's) in the n-th iterate of the mapping 00->0010, 01->011, 10->010, starting with 00; see A289128.

%H Clark Kimberling, <a href="/A289131/b289131.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-2,2).

%F a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 2*a(n-4) + 2*a(n-5) for n >= 6, a(0) = 2, a(1) = 4, a(2) = 7, a(3) = 11, a(4) = 18, a(5) = 28.

%F From _Colin Barker_, Jul 02 2017: (Start)

%F G.f.: (2 + 2*x - 3*x^2 - 2*x^3 + 2*x^4 + 2*x^5) / ((1 - x)^2*(1 + x)*(1 - 2*x^2)).

%F a(n) = -(3*n)/2 + 7*2^(n/2) - 4 for n>0 and even.

%F a(n) = (-3*n + 5*2^((n + 3)/2) - 9) / 2 for n odd.

%F (End)

%t Join[{2}, LinearRecurrence[{1, 3, -3, -2, 2}, {4, 7, 11, 18, 28}, 40]]

%o (PARI) Vec((2 + 2*x - 3*x^2 - 2*x^3 + 2*x^4 + 2*x^5) / ((1 - x)^2*(1 + x)*(1 - 2*x^2)) + O(x^50)) \\ _Colin Barker_, Jul 02 2017

%Y Cf. A289128.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Jun 28 2017

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