A152811 a(n) = 2*(n^2 + 2*n - 2).

2, 12, 26, 44, 66, 92, 122, 156, 194, 236, 282, 332, 386, 444, 506, 572, 642, 716, 794, 876, 962, 1052, 1146, 1244, 1346, 1452, 1562, 1676, 1794, 1916, 2042, 2172, 2306, 2444, 2586, 2732, 2882, 3036, 3194, 3356, 3522, 3692, 3866, 4044, 4226, 4412, 4602, 4796, 4994

Offset: 1

Author

Vincenzo Librandi, Dec 17 2008

Keywords

• nonn
• easy
• less

Comments

Positive numbers k such that 2*k + 12 is a square. [Comment simplified by Zak Seidov, Jan 14 2009]

Sequence gives positive x values of solutions (x,y) to the Diophantine equation 2*x^3 + 12*x^2 = y^2. Corresponding y values are 8*A154560. There are three other solutions: (0,0), (-4,8) and (-6,0).

From a(2) onwards, third subdiagonal of triangle defined in A144562.

Also, nonnegative numbers of the form (m+sqrt(-3))^2 + (m-sqrt(-3))^2. - Bruno Berselli, Mar 13 2015

a(n-1) is the maximum Zagreb index over all maximal 2-degenerate graphs with n vertices. The extremal graphs are 2-stars, so the bound also applies to 2-trees. (The Zagreb index of a graph is the sum of the squares of the degrees over all vertices of the graph.) - Allan Bickle, Apr 11 2024

Examples

			a(4) = 2*(4^2 + 2*4 - 2) = 44 = 2*22 = 2*A028872(5); 2*44^3 + 12*44^2 = 193600 = 440^2 is a square.
The graph K_3 has 3 degree 2 vertices, so a(3-1) = 3*4 = 12.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs, Vol. 0(1) (2024), Article 5.
• Allan Bickle, Zagreb Indices of Maximal k-degenerate Graphs, Australas. J. Combin., Vol. 89(1) (2024), pp. 167-178.
• J. Estes and B. Wei, Sharp bounds of the Zagreb indices of k-trees, J. Comb. Optim., Vol. 27 (2014), pp. 271-291.
• I. Gutman and K. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem., No. 50 (2004), pp. 83-92.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A028872 (n^2-3), A154560 ((n+3)^2*n/2+1), A144562 (triangle T(m,n) = 2m*n+m+n-1).

Cf. A002378, A152811, A371912 (Zagreb indices of maximal k-degenerate graphs).

Programs

Magma

[ 2*(n^2+2*n-2) : n in [1..47] ];

Mathematica

Table[2*n*(n + 2) - 4, {n, 50}] (* Paolo Xausa, Aug 08 2024 *)

PARI

{m=4700; for(n=1, m, if(issquare(2*n^3+12*n^2), print1(n, ",")))}

Formula

G.f.: 2*x*(1 + 3*x - 2*x^2)/(1-x)^3. [corrected by Elmo R. Oliveira, Nov 17 2024]

a(n) = 2*A028872(n+1).

a(n) = a(n-1) + 4*n + 2 for n > 1, a(1)=2.

From Amiram Eldar, Mar 02 2023: (Start)

Sum_{n>=1} 1/a(n) = 1/3 - cot(sqrt(3)*Pi)*Pi/(4*sqrt(3)).

Sum_{n>=1} (-1)^(n+1)/a(n) = -(2 + sqrt(3)*Pi*cosec(sqrt(3)*Pi))/12. (End)

From Elmo R. Oliveira, Nov 17 2024: (Start)

E.g.f.: 2*(exp(x)*(x^2 + 3*x - 2) + 2).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

Extensions

Edited and extended by Klaus Brockhaus, Jan 12 2009