A277983 a(n) = 54*n^2 - 78*n + 36.

36, 12, 96, 288, 588, 996, 1512, 2136, 2868, 3708, 4656, 5712, 6876, 8148, 9528, 11016, 12612, 14316, 16128, 18048, 20076, 22212, 24456, 26808, 29268, 31836, 34512, 37296, 40188, 43188, 46296, 49512, 52836, 56268, 59808, 63456, 67212, 71076, 75048, 79128, 83316

Offset: 0

Author

Emeric Deutsch, Nov 11 2016

Keywords

• nonn
• easy

Comments

For n>=1, a(n) is the second Zagreb index of the triangular grid graph T[n] (see the West reference, p. 390). The second Zagreb index of a simple connected graph is the sum of the degree products d(i)d(j) over all edges ij of the graph.

The M-polynomial of the triangular grid graph T[n] is M(T[n], x, y) = 6*x^2*y^4 + 3*(n-1)*x^4*y^4 +6*(n-2)*x^4*y^6+3*(n-2)*(n-3)*x^6*y^6/2.

References

• D. B. West, Introduction to Graph Theory, 2nd edition, Prentice-Hall, 2001.

Links

• E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
• Eric Weisstein's World of Mathematics, .html">Triangular Grid Graph
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A153792.

Programs

Magma

[54*n^2-78*n+36: n in [0..50]]; // Bruno Berselli, Nov 11 2016

Maple

seq(54*n^2-78*n+36, n=0..40);

Mathematica

Table[54 n^2 - 78 n + 36, {n, 0, 50}] (* Bruno Berselli, Nov 11 2016 *)

PARI

a(n)=54*n^2-78*n+36 \\ Charles R Greathouse IV, Jun 17 2017

Sage

[54*n^2-78*n+36 for n in range(50)] # Bruno Berselli, Nov 11 2016

Formula

O.g.f.: 12*(14*x^2 - 8*x + 3)/(1 - x)^3.

E.g.f.: 6*(9*x^2 - 4*x + 6)*exp(x). - Bruno Berselli, Nov 11 2016

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - Wesley Ivan Hurt, Jan 15 2022

a(n) = 12*A064225(n-1). - R. J. Mathar, Jul 22 2022