OFFSET
1,1
COMMENTS
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.
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
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, 619-627, 2017.
LINKS
Muniru A Asiru, Table of n, a(n) for n = 1..3000
E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = 64*n^2 + 64*n - 32.
G.f.: 32*x*(3+2*x-x^2)/(1-x)^3. - Vincenzo Librandi, Sep 24 2017
EXAMPLE
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.
MAPLE
a := proc (n) options operator, arrow: 64*n^2+64*n-32 end proc: seq(a(n), n = 1 .. 40);
MATHEMATICA
Table[64 n^2 + 64 n - 32, {n, 40}] (* Michael De Vlieger, Sep 23 2017 *)
CoefficientList[Series[32 (3 + 2 x - x^2) / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Sep 24 2017 *)
PROG
(Magma) [64*n^2+64*n-32: n in [1..40]]; // Vincenzo Librandi, Sep 24 2017
(GAP) List([1..50], n->64*n^2+64*n-32); # Muniru A Asiru, May 10 2018
(PARI) a(n) = 64*n^2+64*n-32; \\ Altug Alkan, May 10 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Sep 23 2017
STATUS
approved