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%I #35 Sep 08 2022 08:46:19
%S 96,352,736,1248,1888,2656,3552,4576,5728,7008,8416,9952,11616,13408,
%T 15328,17376,19552,21856,24288,26848,29536,32352,35296,38368,41568,
%U 44896,48352,51936,55648,59488,63456,67552,71776,76128,80608,85216,89952,94816,99808,104928
%N The second Zagreb index of the Aztec diamond AZ(n) (see the Ramanes et al. reference, Theorem 2.2).
%C The second Zagreb index of a simple connected graph g is the sum of the degree products d(i)d(j) over all edges ij of g.
%C The M-polynomial of the Aztec diamond AZ(n) is M(AZ(n); x,y) = 8*x^2*y^3 + 8*(n-1)*x^2*y^4 + 4*x^3*y^4 + 4*(n^2 - 1)*x^4*y^4. - _Emeric Deutsch_, May 10 2018
%D M. Imran and S. Hayat, On computation of topological indices of Aztec diamonds, Sci. Int. (Lahore), 26 (4), 1407-1412, 2014.
%D H. S. Ramanes and R. B. Jummannaver, Computation of Zagreb indices and forgotten index of Aztec diamond, Aryabhatta J. Math. and Informatics, Vol. 09, No. 01, 619-627, 2017.
%H Muniru A Asiru, <a href="/A292345/b292345.txt">Table of n, a(n) for n = 1..3000</a>
%H E. Deutsch and Sandi Klavzar, <a href="http://dx.doi.org/10.22052/ijmc.2015.10106">M-polynomial and degree-based topological indices</a>, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 64*n^2 + 64*n - 32.
%F G.f.: 32*x*(3+2*x-x^2)/(1-x)^3. - _Vincenzo Librandi_, Sep 24 2017
%e a(1) = 96; indeed, the Aztec diamond AZ(1) has 8 edges connecting a vertex of degree 2 with a vertex of degree 3 and 4 edges connecting a vertex of degree 3 with a vertex of degree 4 (see p. 620 of the Ramanes et al. reference); consequently, a(1) = 8*2*3 + 4*3*4 = 48 + 48 = 96.
%p a := proc (n) options operator, arrow: 64*n^2+64*n-32 end proc: seq(a(n), n = 1 .. 40);
%t Table[64 n^2 + 64 n - 32, {n, 40}] (* _Michael De Vlieger_, Sep 23 2017 *)
%t CoefficientList[Series[32 (3 + 2 x - x^2) / (1 - x)^3, {x, 0, 40}], x] (* _Vincenzo Librandi_, Sep 24 2017 *)
%o (Magma) [64*n^2+64*n-32: n in [1..40]]; // _Vincenzo Librandi_, Sep 24 2017
%o (GAP) List([1..50],n->64*n^2+64*n-32); # _Muniru A Asiru_, May 10 2018
%o (PARI) a(n) = 64*n^2+64*n-32; \\ _Altug Alkan_, May 10 2018
%Y Cf. A292344.
%K nonn,easy
%O 1,1
%A _Emeric Deutsch_, Sep 23 2017