A292344 The first Zagreb index of the Aztec diamond AZ(n) (see the Ramanes et al. reference, Theorem 2.1).

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

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 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..2000

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) = 32*n^2 + 48*n - 12.

G.f.: 4*x*(17+2*x-3*x^2)/(1-x)^3. - Vincenzo Librandi, Sep 24 2017

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*n-12 end proc: seq(a(n), n = 1 .. 40);

Mathematica

Table[32 n^2 + 48 n - 12, {n, 40}] (* Michael De Vlieger, Sep 23 2017 *)
CoefficientList[Series[4 (17 + 2 x - 3 x^2) / (1-x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Sep 24 2017 *)

Prog

(Magma) [32*n^2+48*n-12: n in [1..40]]; // Vincenzo Librandi, Sep 24 2017
(GAP) List([1..50], n->32*n^2+48*n-12); # Muniru A Asiru, May 10 2018
(PARI) a(n) = 32*n^2+48*n-12; \\ Altug Alkan, May 10 2018

Crossrefs

Cf. A292345.

Sequence in context: A044400 A044781 A317807 * A189810 A200198 A237745

Adjacent sequences: A292341 A292342 A292343 * A292345 A292346 A292347

Keyword

nonn,easy

Author

Emeric Deutsch, Sep 23 2017

Status

approved