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%I #32 Sep 08 2022 08:45:31
%S 1,5,11,21,39,73,139,269,527,1041,2067,4117,8215,16409,32795,65565,
%T 131103,262177,524323,1048613,2097191,4194345,8388651,16777261,
%U 33554479,67108913,134217779,268435509,536870967,1073741881,2147483707,4294967357,8589934655,17179869249
%N a(n) = 2^(n+1) + 2*n - 1.
%C Row sums of triangle A131897.
%C Binomial transform of (1, 4, 2, 2, 2, ...).
%C a(n), n > 0, is the number of maximal subsemigroups of the Motzkin monoid of degree n + 1. - _James Mitchell_ and _Wilf A. Wilson_, Jul 21 2017
%H Vincenzo Librandi, <a href="/A131898/b131898.txt">Table of n, a(n) for n = 0..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. [From _James Mitchell_ and _Wilf A. Wilson_, Jul 21 2017]
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).
%F G.f.: ( -1-x+4*x^2 ) / ( (2*x-1)*(x-1)^2 ). - _R. J. Mathar_, Jul 03 2011
%F a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3). - _Vincenzo Librandi_, Jul 05 2012
%e a(3) = 21 = sum of row 3 terms of triangle A131897: (11 + 4 + 2 + 4).
%e a(3) = 21 = (1, 3, 3, 1) dot (1, 4, 2, 2) = (1 + 12 + 6 + 2).
%t CoefficientList[Series[(-1-x+4*x^2)/((2*x-1)*(x-1)^2),{x,0,40}],x] (* _Vincenzo Librandi_, Jul 05 2012 *)
%o (Magma) I:=[1, 5, 11]; [n le 3 select I[n] else 4*Self(n-1)-5*Self(n-2)+2*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Jul 05 2012
%Y Cf. A131897.
%K nonn,easy
%O 0,2
%A _Gary W. Adamson_, Jul 25 2007
%E New definition by _R. J. Mathar_, Jul 03 2011