OFFSET
1,2
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).
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)
MAPLE
seq((2*k)^2-4, k=1..46);
MATHEMATICA
a[n_] := (2*n)^2 - 4; Array[a, 50] (* Amiram Eldar, Dec 10 2022 *)
PROG
(PARI) a(n)=(2*n)^2-4 \\ Charles R Greathouse IV, Jun 16 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Zerinvary Lajos, Jan 23 2008
STATUS
approved