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%I #33 Sep 08 2022 08:46:19
%S 68,212,420,692,1028,1428,1892,2420,3012,3668,4388,5172,6020,6932,
%T 7908,8948,10052,11220,12452,13748,15108,16532,18020,19572,21188,
%U 22868,24612,26420,28292,30228,32228,34292,36420,38612,40868,43188,45572,48020,50532,53108
%N The first Zagreb index of the Aztec diamond AZ(n) (see the Ramanes et al. reference, Theorem 2.1).
%C 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.
%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="/A292344/b292344.txt">Table of n, a(n) for n = 1..2000</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) = 32*n^2 + 48*n - 12.
%F G.f.: 4*x*(17+2*x-3*x^2)/(1-x)^3. - _Vincenzo Librandi_, Sep 24 2017
%e a(1) = 68; indeed, the Aztec diamond AZ(1) has 4 vertices of degree 2, 4 vertices of degree 3, and 1 vertex of degree 4 (see p. 1409 of the Imran et al. reference); consequently, a(1) = 4*2^2 + 4*3^2 + 1*4^2 = 16 + 36 + 16 = 68.
%p a:= proc(n) options operator, arrow: 32*n^2+48*n-12 end proc: seq(a(n), n = 1 .. 40);
%t Table[32 n^2 + 48 n - 12, {n, 40}] (* _Michael De Vlieger_, Sep 23 2017 *)
%t CoefficientList[Series[4 (17 + 2 x - 3 x^2) / (1-x)^3, {x, 0, 40}], x] (* _Vincenzo Librandi_, Sep 24 2017 *)
%o (Magma) [32*n^2+48*n-12: n in [1..40]]; // _Vincenzo Librandi_, Sep 24 2017
%o (GAP) List([1..50],n->32*n^2+48*n-12); # _Muniru A Asiru_, May 10 2018
%o (PARI) a(n) = 32*n^2+48*n-12; \\ _Altug Alkan_, May 10 2018
%Y Cf. A292345.
%K nonn,easy
%O 1,1
%A _Emeric Deutsch_, Sep 23 2017
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