A152811 - OEIS

%I #52 Nov 17 2024 17:09:54

%S 2,12,26,44,66,92,122,156,194,236,282,332,386,444,506,572,642,716,794,

%T 876,962,1052,1146,1244,1346,1452,1562,1676,1794,1916,2042,2172,2306,

%U 2444,2586,2732,2882,3036,3194,3356,3522,3692,3866,4044,4226,4412,4602,4796,4994

%N a(n) = 2*(n^2 + 2*n - 2).

%C Positive numbers k such that 2*k + 12 is a square. [Comment simplified by _Zak Seidov_, Jan 14 2009]

%C 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).

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

%C Also, nonnegative numbers of the form (m+sqrt(-3))^2 + (m-sqrt(-3))^2. - _Bruno Berselli_, Mar 13 2015

%C 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

%H Vincenzo Librandi, <a href="/A152811/b152811.txt">Table of n, a(n) for n = 1..10000</a>

%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, Vol. 0(1) (2024), Article 5.

%H Allan Bickle, <a href="https://ajc.maths.uq.edu.au/pdf/89/ajc_v89_p167.pdf">Zagreb Indices of Maximal k-degenerate Graphs</a>, Australas. J. Combin., Vol. 89(1) (2024), pp. 167-178.

%H J. Estes and B. Wei, <a href="https://doi.org/10.1007/s10878-012-9515-6">Sharp bounds of the Zagreb indices of k-trees</a>, J. Comb. Optim., Vol. 27 (2014), pp. 271-291.

%H I. Gutman and K. Das, <a href="https://match.pmf.kg.ac.rs/electronic_versions/Match50/match50_83-92.pdf">The first Zagreb index 30 years after</a>, MATCH Commun. Math. Comput. Chem., No. 50 (2004), pp. 83-92.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

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

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

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

%F From _Amiram Eldar_, Mar 02 2023: (Start)

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

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

%F From _Elmo R. Oliveira_, Nov 17 2024: (Start)

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

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

%e 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.

%e The graph K_3 has 3 degree 2 vertices, so a(3-1) = 3*4 = 12.

%t Table[2*n*(n + 2) - 4, {n, 50}] (* _Paolo Xausa_, Aug 08 2024 *)

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

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

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

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

%K nonn,easy,less

%O 1,1

%A _Vincenzo Librandi_, Dec 17 2008

%E Edited and extended by _Klaus Brockhaus_, Jan 12 2009