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A292343
The PI index of the Aztec diamond AZ(n) (see the Imran et al. reference).
1
108, 888, 3268, 8560, 18460, 35048, 60788, 98528, 151500, 223320, 317988, 439888, 593788, 784840, 1018580, 1300928, 1638188, 2037048, 2504580, 3048240, 3675868, 4395688, 5216308, 6146720, 7196300, 8374808, 9692388, 11159568, 12787260, 14586760, 16569748
OFFSET
1,1
REFERENCES
M. Imran and S. Hayat, On computation of topological indices of Aztec diamonds, Sci. Int. (Lahore), 26 (4), 1407-1412, 2014.
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.
FORMULA
a(n) = (4/3)*n*(12*n^3 + 44*n^2 + 36*n - 11).
G.f.: 4*x*(27 + 87*x - 23*x^2 + 5*x^3)/(1 - x)^5. - Vincenzo Librandi, Sep 24 2017
EXAMPLE
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.
MAPLE
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);
MATHEMATICA
Table[(4/3) n (12 n^3 + 44 n^2 + 36 n - 11), {n, 31}] (* Michael De Vlieger, Sep 23 2017 *)
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 *)
PROG
(Magma) [(4/3)*n*(12*n^3+44*n^2+36*n-11): n in [1..40]]; // Vincenzo Librandi, Sep 24 2017
CROSSREFS
Sequence in context: A202194 A333757 A203373 * A228174 A184201 A101213
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Sep 23 2017
STATUS
approved