A292345 The second Zagreb index of the Aztec diamond AZ(n) (see the Ramanes et al. reference, Theorem 2.2).

96, 352, 736, 1248, 1888, 2656, 3552, 4576, 5728, 7008, 8416, 9952, 11616, 13408, 15328, 17376, 19552, 21856, 24288, 26848, 29536, 32352, 35296, 38368, 41568, 44896, 48352, 51936, 55648, 59488, 63456, 67552, 71776, 76128, 80608, 85216, 89952, 94816, 99808, 104928

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

Cf. A292344.

Sequence in context: A304830 A301459 A220540 * A084048 A167983 A248457

Adjacent sequences: A292342 A292343 A292344 * A292346 A292347 A292348

Keyword

nonn,easy

Author

Emeric Deutsch, Sep 23 2017

Status

approved