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

Author

Paul Barry, Feb 09 2005

Keywords

• easy
• nonn

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).

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).

Programs

Magma

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

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 *)

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