A113127 - OEIS

%I #50 Feb 18 2024 02:09:33

%S 1,3,6,10,14,18,22,26,30,34,38,42,46,50,54,58,62,66,70,74,78,82,86,90,

%T 94,98,102,106,110,114,118,122,126,130,134,138,142,146,150,154,158,

%U 162,166,170,174,178,182,186,190,194,198,202,206,210,214,218,222,226,230

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

%C Row sums of number triangle A113126.

%C Equals binomial transform of [1, 2, 1, 0, -1, 2, -3, 4, -5, ...]. - _Gary W. Adamson_, Feb 14 2009

%C From 6 on the same as A016825. - _R. J. Mathar_, Jul 21 2013

%C The size of a maximal 4-degenerate graph of order n-2 (this class includes 4-trees). - _Allan Bickle_, Nov 14 2021

%C 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

%H Vincenzo Librandi, <a href="/A113127/b113127.txt">Table of n, a(n) for n = 0..5000</a>

%H Allan Bickle, <a href="https://doi.org/10.7151/dmgt.1637">Structural results on maximal k-degenerate graphs</a>, Discuss. Math. Graph Theory 32 4 (2012), 659-676.

%H Allan Bickle, <a href="https://doi.org/10.20429/tag.2024.000105">A Survey of Maximal k-degenerate Graphs and k-Trees</a>, Theory and Applications of Graphs 0 1 (2024) Article 5.

%H D. R. Lick and A. T. White, <a href="https://doi.org/10.4153/CJM-1970-125-1">k-degenerate graphs</a>, Canad. J. Math. 22 (1970), 1082-1096.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

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

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

%F Row sums of triangle A131034. - _Gary W. Adamson_, Jun 10 2007

%F 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

%F a(n) = A111284(n+1) for n >= 2. - _Georg Fischer_, Nov 02 2018

%F a(n) = 4*(n+2) - 10 for n >= 2. - _Allan Bickle_, Nov 14 2021

%t CoefficientList[Series[(1 + x + x^2 + x^3) / (1 - x)^2, {x, 0, 100}], x] (* _Vincenzo Librandi_, Nov 03 2018 *)

%t LinearRecurrence[{2,-1},{1,3,6,10},60] (* _Harvey P. Dale_, Jul 08 2019 *)

%o (PARI) x='x+O('x^66); Vec((1+x+x^2+x^3)/(1-x)^2) \\ _Joerg Arndt_, May 06 2013

%o (Magma) [4*n-2+2*Binomial(0, n)+Binomial(1, n): n in [0..80]]; // _Vincenzo Librandi_, Nov 03 2018

%Y Cf. A016825, A073760, A111284, A131034.

%Y a(n) - a(n-1) = A158411(n+1). - _Jaume Oliver Lafont_, Mar 27 2009

%K easy,nonn

%O 0,2

%A _Paul Barry_, Oct 14 2005