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The PI index of the Aztec diamond AZ(n) (see the Imran et al. reference).
1

%I #15 Sep 08 2022 08:46:19

%S 108,888,3268,8560,18460,35048,60788,98528,151500,223320,317988,

%T 439888,593788,784840,1018580,1300928,1638188,2037048,2504580,3048240,

%U 3675868,4395688,5216308,6146720,7196300,8374808,9692388,11159568,12787260,14586760,16569748

%N The PI index of the Aztec diamond AZ(n) (see the Imran et al. reference).

%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, 2017.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = (4/3)*n*(12*n^3 + 44*n^2 + 36*n - 11).

%F G.f.: 4*x*(27 + 87*x - 23*x^2 + 5*x^3)/(1 - x)^5. - _Vincenzo Librandi_, Sep 24 2017

%e a(1) = 108; indeed, the Aztec diamond AZ(1) has 12 edges and 9 vertices (see p. 1409 of the Imran et al. reference); for each edge uv, none of the 9 vertices is equidistant from u and v; consequently, a(1) = 12*9 = 108.

%p a := proc (n) options operator, arrow: (4/3)*n*(12*n^3+44*n^2+36*n-11) end proc: seq(a(n), n = 1 .. 40);

%t Table[(4/3) n (12 n^3 + 44 n^2 + 36 n - 11), {n, 31}] (* _Michael De Vlieger_, Sep 23 2017 *)

%t CoefficientList[Series[4 (27 + 87 x - 23 x^2 + 5 x^3) / (1 - x)^5, {x, 0, 40}], x] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {108, 888, 3268, 8560, 18460}, 40] (* _Vincenzo Librandi_, Sep 24 2017 *)

%o (Magma) [(4/3)*n*(12*n^3+44*n^2+36*n-11): n in [1..40]]; // _Vincenzo Librandi_, Sep 24 2017

%K nonn,easy

%O 1,1

%A _Emeric Deutsch_, Sep 23 2017