%I #23 Feb 14 2021 12:34:37

%S 1,2,5,9,13,19,27,39,57,85,129,199,311,491,781,1249,2005,3227,5203,

%T 8399,13569,21933,35465,57359,92783,150099,242837,392889,635677,

%U 1028515,1664139,2692599,4356681,7049221,11405841,18454999,29860775,48315707,78176413

%N The number of maximal subsemigroups of the Jones monoid on the set [1..n].

%C a(2n) is the number of maximal subsemigroups of the planar partition monoid of degree n.

%H Wilf A. Wilson, <a href="/A290140/b290140.txt">Table of n, a(n) for n = 1..1000</a>

%H James East, Jitender Kumar, James D. Mitchell, and Wilf A. Wilson, <a href="https://arxiv.org/abs/1706.04967">Maximal subsemigroups of finite transformation and partition monoids</a>, arXiv:1706.04967 [math.GR], 2017.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).

%F a(n) = 2 * A000045(n - 1) + 2n - 3, n > 2.

%F From _Colin Barker_, Jul 21 2017: (Start)

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

%F a(n) = -5 + (2^(-n)*((1-sqrt(5))^n*(1+sqrt(5)) + (-1+sqrt(5))*(1+sqrt(5))^n)) / sqrt(5) + 2*(1+n) for n>2.

%F a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) for n>6.

%F (End)

%t {1, 2}~Join~Table[2 Fibonacci[n - 1] + 2 n - 3, {n, 3, 39}] (* _Michael De Vlieger_, Jul 21 2017 *)

%o (PARI) Vec(x*(1 + x)*(1 - 2*x + 3*x^2 - 4*x^3 + x^4) / ((1 - x)^2*(1 - x - x^2)) + O(x^50)) \\ _Colin Barker_, Jul 21 2017

%Y Cf. A000045.

%K nonn,easy

%O 1,2

%A _James Mitchell_ and _Wilf A. Wilson_, Jul 21 2017