A134582 a(n) = (2*n)^2 - 4.

0, 12, 32, 60, 96, 140, 192, 252, 320, 396, 480, 572, 672, 780, 896, 1020, 1152, 1292, 1440, 1596, 1760, 1932, 2112, 2300, 2496, 2700, 2912, 3132, 3360, 3596, 3840, 4092, 4352, 4620, 4896, 5180, 5472, 5772, 6080, 6396, 6720, 7052, 7392, 7740, 8096, 8460

Offset: 1

Author

Zerinvary Lajos, Jan 23 2008

Keywords

• nonn
• easy

Comments

a(n) is the first Zagreb index of the friendship graph F[n-1]. The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph. The friendship graph (or Dutch windmill graph) F[n] can be constructed by joining n copies of the cycle graph C[3] with a common vertex. a(3) = 32. Indeed, the friendship graph F[2] has 2 edges with end-point degrees 2,2 and 4 edges with end-point degrees 2,4. Then the first Zagreb index is 2*4 + 4*6 = 32. - Emeric Deutsch, Nov 09 2016

a(n) is also the number of edges of the Aztec diamond AZ(n-1), (n>=2), (see Lemma 2.2 of the Imran et al. paper). - Emeric Deutsch, Sep 23 2017

For n >= 2, the continued fraction expansion of sqrt(a(n)) is [2n-1; {1, n-2, 1, 4n-2}]. For n=2, this collapses to [3; {2, 6}]. - Magus K. Chu, Nov 14 2022

References

• M. Imran and S. Hayat, On computation of topological indices of Aztec diamonds, Sci. Int. (Lahore), 26 (4), 1407-1412, 2014.

Links

• R. E. Borcherds, E. Freitag, and R. Weissauer, A Siegel cusp form of degree 12 and weight 12, arXiv:math/9805132 [math.AG], 1998, row A_2 page 6.
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
• Wikipedia, Friendship graph.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A005563.

Programs

Maple

seq((2*k)^2-4, k=1..46);

Mathematica

a[n_] := (2*n)^2 - 4; Array[a, 50] (* Amiram Eldar, Dec 10 2022 *)

PARI

a(n)=(2*n)^2-4 \\ Charles R Greathouse IV, Jun 16 2017

Formula

From R. J. Mathar, Jan 24 2008: (Start)

O.g.f.: 4 - 12/(-1+x)^2 - 8/(-1+x)^3.

a(n) = 4*A005563(n-1). (End)

a(n) = a(n-1) + 8*n - 4 (with a(1)=0). - Vincenzo Librandi, Nov 23 2010

From Amiram Eldar, Dec 10 2022: (Start)

Sum_{n>=2} 1/a(n) = 3/16.

Sum_{n>=2} (-1)^n/a(n) = 1/16. (End)