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%I #19 Sep 08 2022 08:46:19
%S 204,748,1548,2604,3916,5484,7308,9388,11724,14316,17164,20268,23628,
%T 27244,31116,35244,39628,44268,49164,54316,59724,65388,71308,77484,
%U 83916,90604,97548,104748,112204,119916,127884,136108,144588,153324,162316,171564
%N The forgotten topological index of the Aztec diamond AZ(n) (see the Ramanes et al. reference, Theorem 2.1).
%C The forgotten topological index of a simple connected graph is the sum of the cubes of its vertex degrees.
%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 B. Furtula and I. Gutman, <a href="https://doi.org/10.1007/s10910-015-0480-z">A forgotten topological index</a>, J. Math. Chem. 53 (4), 1184-1190, 2015.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 128*n^2 + 160*n - 84.
%F G.f.: 4*x*(51+34*x-21*x^2)/(1-x)^3. - _Vincenzo Librandi_, Sep 24 2017
%e a(1) = 204; indeed, the Aztec diamond AZ(1) has four vertices of degree 2, four vertices of degree 3, and one vertex of degree 4 (see p. 620 of the Ramanes et al. reference); consequently, a(1) = 4*8 + 4*27 + 1*64 = 32 + 108 + 64 = 204.
%p a := proc (n) options operator, arrow: 128*n^2+160*n-84 end proc: seq(a(n), n = 1 .. 40);
%t Table[128 n^2 + 160 n - 84, {n, 36}] (* _Michael De Vlieger_, Sep 23 2017 *)
%t CoefficientList[Series[4 (51 + 34 x - 21 x^2) / (1-x)^3, {x, 0, 40}], x] (* _Vincenzo Librandi_, Sep 24 2017 *)
%o (Magma) [128*n^2+160*n-84: n in [1..40]]; // _Vincenzo Librandi_, Sep 24 2017
%Y Cf. A292343, A292344, A292345.
%K nonn,easy
%O 1,1
%A _Emeric Deutsch_, Sep 23 2017
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