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A292345 The second Zagreb index of the Aztec diamond AZ(n) (see the Ramanes et al. reference, Theorem 2.2). 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 11 03:07 EDT 2020. Contains 336421 sequences. (Running on oeis4.)