OFFSET
1,1
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
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).
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)
EXAMPLE
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.
MATHEMATICA
Table[2*n*(n + 2) - 4, {n, 50}] (* Paolo Xausa, Aug 08 2024 *)
PROG
(Magma) [ 2*(n^2+2*n-2) : n in [1..47] ];
(PARI) {m=4700; for(n=1, m, if(issquare(2*n^3+12*n^2), print1(n, ", ")))}
CROSSREFS
KEYWORD
nonn,easy,less
AUTHOR
Vincenzo Librandi, Dec 17 2008
EXTENSIONS
Edited and extended by Klaus Brockhaus, Jan 12 2009
STATUS
approved