A131898 a(n) = 2^(n+1) + 2*n - 1.

1, 5, 11, 21, 39, 73, 139, 269, 527, 1041, 2067, 4117, 8215, 16409, 32795, 65565, 131103, 262177, 524323, 1048613, 2097191, 4194345, 8388651, 16777261, 33554479, 67108913, 134217779, 268435509, 536870967, 1073741881, 2147483707, 4294967357, 8589934655, 17179869249

Offset: 0

Author

Gary W. Adamson, Jul 25 2007

Keywords

• nonn
• easy

Comments

Row sums of triangle A131897.

Binomial transform of (1, 4, 2, 2, 2, ...).

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

Examples

			a(3) = 21 = sum of row 3 terms of triangle A131897: (11 + 4 + 2 + 4).
a(3) = 21 = (1, 3, 3, 1) dot (1, 4, 2, 2) = (1 + 12 + 6 + 2).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• James East, Jitender Kumar, James D. Mitchell, and Wilf A. Wilson, Maximal subsemigroups of finite transformation and partition monoids, arXiv:1706.04967 [math.GR], 2017. [From _James Mitchell_ and _Wilf A. Wilson_, Jul 21 2017]
• Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Crossrefs

Cf. A131897.

Programs

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

Mathematica

CoefficientList[Series[(-1-x+4*x^2)/((2*x-1)*(x-1)^2),{x,0,40}],x] (* Vincenzo Librandi, Jul 05 2012 *)

Formula

G.f.: (-1-x+4*x^2)/((2*x-1)*(x-1)^2). - R. J. Mathar, Jul 03 2011

a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Vincenzo Librandi, Jul 05 2012

E.g.f.: exp(x)*(2*(exp(x) + x) - 1). - Elmo R. Oliveira, Mar 08 2025

Extensions

New definition by R. J. Mathar, Jul 03 2011