A113127 Expansion of (1 + x + x^2 + x^3)/(1-x)^2.

1, 3, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 226, 230

Offset

0,2

Comments

Row sums of number triangle A113126.

Equals binomial transform of [1, 2, 1, 0, -1, 2, -3, 4, -5, ...]. - Gary W. Adamson, Feb 14 2009

From 6 on the same as A016825. - R. J. Mathar, Jul 21 2013

The size of a maximal 4-degenerate graph of order n-2 (this class includes 4-trees). - Allan Bickle, Nov 14 2021

Maximum size of an apex graph of order n-2 (an apex graph can be made planar by deleting a single vertex). - Allan Bickle, Nov 14 2021

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

Allan Bickle, Structural results on maximal k-degenerate graphs, Discuss. Math. Graph Theory 32 4 (2012), 659-676.

Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.

D. R. Lick and A. T. White, k-degenerate graphs, Canad. J. Math. 22 (1970), 1082-1096.

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

Formula

a(n) = 4*n - 2 + 2*binomial(0, n) + binomial(1, n);

a(n) = binomial(n+1, n) + binomial(n, n-1) + binomial(n-1, n-2) + binomial(n-2, n-3).

Row sums of triangle A131034. - Gary W. Adamson, Jun 10 2007

G.f.: (x^2-1)/Q(0), where Q(k)= 4*x - 1 + x*k - x*(x-1)*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 05 2013

a(n) = A111284(n+1) for n >= 2. - Georg Fischer, Nov 02 2018

a(n) = 4*(n+2) - 10 for n >= 2. - Allan Bickle, Nov 14 2021

Mathematica

CoefficientList[Series[(1 + x + x^2 + x^3) / (1 - x)^2, {x, 0, 100}], x] (* Vincenzo Librandi, Nov 03 2018 *)
LinearRecurrence[{2, -1}, {1, 3, 6, 10}, 60] (* Harvey P. Dale, Jul 08 2019 *)

Prog

(PARI) x='x+O('x^66); Vec((1+x+x^2+x^3)/(1-x)^2) \\ Joerg Arndt, May 06 2013
(Magma) [4*n-2+2*Binomial(0, n)+Binomial(1, n): n in [0..80]]; // Vincenzo Librandi, Nov 03 2018

Crossrefs

Cf. A016825, A073760, A111284, A131034.

a(n) - a(n-1) = A158411(n+1). - Jaume Oliver Lafont, Mar 27 2009

Sequence in context: A348675 A338074 A310064 * A145913 A130246 A167381

Adjacent sequences: A113124 A113125 A113126 * A113128 A113129 A113130

Keyword

easy,nonn

Author

Paul Barry, Oct 14 2005

Status

approved