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%I #55 Dec 10 2022 10:46:11
%S 0,12,32,60,96,140,192,252,320,396,480,572,672,780,896,1020,1152,1292,
%T 1440,1596,1760,1932,2112,2300,2496,2700,2912,3132,3360,3596,3840,
%U 4092,4352,4620,4896,5180,5472,5772,6080,6396,6720,7052,7392,7740,8096,8460
%N a(n) = (2*n)^2 - 4.
%C 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
%C 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
%C 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
%D M. Imran and S. Hayat, On computation of topological indices of Aztec diamonds, Sci. Int. (Lahore), 26 (4), 1407-1412, 2014.
%H R. E. Borcherds, E. Freitag, and R. Weissauer, <a href="http://arXiv.org/abs/math/9805132">A Siegel cusp form of degree 12 and weight 12</a>, arXiv:math/9805132 [math.AG], 1998, row A_2 page 6.
%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Friendship_graph">Friendship graph</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F From _R. J. Mathar_, Jan 24 2008: (Start)
%F O.g.f.: 4 - 12/(-1+x)^2 - 8/(-1+x)^3.
%F a(n) = 4*A005563(n-1). (End)
%F a(n) = a(n-1) + 8*n - 4 (with a(1)=0). - _Vincenzo Librandi_, Nov 23 2010
%F From _Amiram Eldar_, Dec 10 2022: (Start)
%F Sum_{n>=2} 1/a(n) = 3/16.
%F Sum_{n>=2} (-1)^n/a(n) = 1/16. (End)
%p seq((2*k)^2-4, k=1..46);
%t a[n_] := (2*n)^2 - 4; Array[a, 50] (* _Amiram Eldar_, Dec 10 2022 *)
%o (PARI) a(n)=(2*n)^2-4 \\ _Charles R Greathouse IV_, Jun 16 2017
%Y Cf. A005563.
%K nonn,easy
%O 1,2
%A _Zerinvary Lajos_, Jan 23 2008
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