

A134582


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


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
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OFFSET

1,2


COMMENTS

a(n) is the first Zagreb index of the friendship graph F[n1]. 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 endpoint degrees 2,2 and 4 edges with endpoint 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(n1), (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 [2n1; {1, n2, 1, 4n2}]. 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), 14071412, 2014.


LINKS



FORMULA

O.g.f.: 4  12/(1+x)^2  8/(1+x)^3.
Sum_{n>=2} 1/a(n) = 3/16.
Sum_{n>=2} (1)^n/a(n) = 1/16. (End)


MAPLE

seq((2*k)^24, k=1..46);


MATHEMATICA

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


PROG



CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



