

A292344


The first Zagreb index of the Aztec diamond AZ(n) (see the Ramanes et al. reference, Theorem 2.1).


3



68, 212, 420, 692, 1028, 1428, 1892, 2420, 3012, 3668, 4388, 5172, 6020, 6932, 7908, 8948, 10052, 11220, 12452, 13748, 15108, 16532, 18020, 19572, 21188, 22868, 24612, 26420, 28292, 30228, 32228, 34292, 36420, 38612, 40868, 43188, 45572, 48020, 50532, 53108
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OFFSET

1,1


COMMENTS

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.
The Mpolynomial of the Aztec diamond AZ(n) is M(AZ(n);x,y) = 8*x^2*y^3 + 8*(n1)*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), 14071412, 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, 619627, 2017.


LINKS



FORMULA

a(n) = 32*n^2 + 48*n  12.


EXAMPLE

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.


MAPLE

a:= proc(n) options operator, arrow: 32*n^2+48*n12 end proc: seq(a(n), n = 1 .. 40);


MATHEMATICA

CoefficientList[Series[4 (17 + 2 x  3 x^2) / (1x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Sep 24 2017 *)


PROG

(PARI) a(n) = 32*n^2+48*n12; \\ Altug Alkan, May 10 2018


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



