A103517 Expansion of (1+2*x-x^2)/(1-x)^2.

1, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126

Offset

0,2

Comments

Row sums of A103516.

Also the number of maximal and maximum cliques in the (n+1) X (n+1) rook graph. - Eric W. Weisstein, Sep 14 2017

Also the number of maximal and maximum independent vertex sets in the (n+1) X (n+1) rook complement graph. - Eric W. Weisstein, Sep 14 2017

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Maximal Clique

Eric Weisstein's World of Mathematics, Maximum Clique

Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set

Eric Weisstein's World of Mathematics, Maximum Independent Vertex Set

Eric Weisstein's World of Mathematics, Rook Complement Graph

Eric Weisstein's World of Mathematics, Rook Graph

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

a(n) = 2*n + 2 - 0^n.

a(n) = Sum_{k=0..n} 0^(k(n-k))*(n+1).

Equals binomial transform of [1, 3, -1, 1, -1, 1, ...]. - Gary W. Adamson, Apr 23 2008

a(n) = 2*a(n-1) - a(n-2) for n > 2. - Eric W. Weisstein, Sep 14 2017

G.f.: (1 + 2*x - x^2)/(-1 + x)^2. - Eric W. Weisstein, Sep 14 2017

Maple

a := n -> 2*(n + 1) - 0^n: seq(a(n), n = 0..62); # Peter Luschny, May 12 2023

Mathematica

CoefficientList[Series[(-z^2 + 2*z + 1)/(z - 1)^2, {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 10 2011 *)
CoefficientList[Series[2/(x - 1)^2 - 1, {x, 0, 62}], x] (* Robert G. Wilson v, Jan 29 2015 *)
Join[{1}, 2 Range[2, 20]] (* Eric W. Weisstein, Sep 14 2017 *)
Join[{1}, LinearRecurrence[{2, -1}, {4, 6}, 20]] (* Eric W. Weisstein, Sep 14 2017 *)

Prog

(Magma) [1] cat [2*n+2 : n in [1..60]]; // Wesley Ivan Hurt, Dec 07 2016

Crossrefs

Cf. A103516.

Essentially the same as A004277, A005843, A051755, and A076032. - R. J. Mathar, Jul 31 2010

Cf. A272651 (for which this sequence is a conjectured continuation for large n).

Sequence in context: A175074 A063127 A272651 * A163300 A193175 A093161

Adjacent sequences: A103514 A103515 A103516 * A103518 A103519 A103520

Keyword

easy,nonn

Author

Paul Barry, Feb 09 2005

Status

approved