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A292344
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The first Zagreb index of the Aztec diamond AZ(n) (see the Ramanes et al. reference, Theorem 2.1).
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3
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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
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1,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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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).
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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a:= proc(n) options operator, arrow: 32*n^2+48*n-12 end proc: seq(a(n), n = 1 .. 40);
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A292345.
Sequence in context: A044400 A044781 A317807 * A189810 A200198 A237745
Adjacent sequences: A292341 A292342 A292343 * A292345 A292346 A292347
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KEYWORD
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nonn,easy
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AUTHOR
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Emeric Deutsch, Sep 23 2017
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STATUS
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approved
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