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

0,2

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

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).

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

Example

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).

Mathematica

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

Prog

(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

Crossrefs

Cf. A131897.

Sequence in context: A163787 A166863 A163704 * A168642 A357750 A234597

Adjacent sequences: A131895 A131896 A131897 * A131899 A131900 A131901

Keyword

nonn,easy,changed

Author

Gary W. Adamson, Jul 25 2007

Extensions

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

Status

approved