A131034 -id:A131034 - cp's OEIS frontend

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

Paul Barry, Oct 14 2005

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

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).

Cf. A016825, A073760, A111284, A131034.

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

[4*n-2+2*Binomial(0, n)+Binomial(1, n): n in [0..80]]; // Vincenzo Librandi, Nov 03 2018

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 *)

x='x+O('x^66); Vec((1+x+x^2+x^3)/(1-x)^2) \\ Joerg Arndt, May 06 2013

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

A131035 A051340 * A129686 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 5, 2, 1, 1, 6, 2, 2, 1, 1, 7, 2, 2, 2, 1, 1, 8, 2, 2, 2, 2, 1, 1, 9, 2, 2, 2, 2, 2, 1, 1, 10, 2, 2, 2, 2, 2, 2, 1, 1

Offset: 1

Gary W. Adamson, Jun 10 2007

Row sums give A008486.

			First few rows of the triangle:
   1;
   2, 1;
   4, 1, 1;
   5, 2, 1, 1;
   6, 2, 2, 1, 1;
   7, 2, 2, 2, 1, 1;
   8, 2, 2, 2, 2, 1, 1;
   9, 2, 2, 2, 2, 2, 1, 1;
  10, 2, 2, 2, 2, 2, 2, 1, 1;
  ...

Cf. A051340, A129686, A008486, A131034.

cp's OEIS Frontend

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

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