A304833 - OEIS

%I #17 May 22 2018 08:14:01

%S 12,65,124,189,260,337,420,509,604,705,812,925,1044,1169,1300,1437,

%T 1580,1729,1884,2045,2212,2385,2564,2749,2940,3137,3340,3549,3764,

%U 3985,4212,4445,4684,4929,5180,5437,5700,5969,6244,6525,6812,7105,7404,7709,8020,8337,8660,8989,9324,9665,10012,10365,10724,11089

%N a(n) = 3*n^2 + 38*n - 76 (n>=2).

%C For n>=3, a(n) is the second Zagreb index of the Mycielskian of  the path graph P[n]. For the Mycielskian, see p. 205 of the West reference and/or the Wikipedia link.

%C 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.

%C For n>=3 the M-polynomial of the considered Mycielskian is 2*x^2*y^3 + 4*x^2*y^4 + 2*x^2*y^n + 2*(n-3)*x^3*y^4 + (n-2)*x^3*y^n +(n-3)*x^4*y^4.

%D D. B. West, Introduction to  Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001.

%H Colin Barker, <a href="/A304833/b304833.txt">Table of n, a(n) for n = 2..1000</a>

%H E. Deutsch and Sandi Klavzar, <a href="http://dx.doi.org/10.22052/ijmc.2015.10106">M-polynomial and degree-based topological indices</a>, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Mycielskian">Mycielskian</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F From _Colin Barker_, May 21 2018: (Start)

%F G.f.: x^2*(12 + 29*x - 35*x^2) / (1 - x)^3.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>4.

%F (End)

%p seq(3*n^2+38*n-76, n = 2 .. 55);

%o (PARI) a(n) = 3*n^2 + 38*n - 76 \\ _Felix Fröhlich_, May 20 2018

%o (PARI) Vec(x^2*(12 + 29*x - 35*x^2) / (1 - x)^3 + O(x^40)) \\ _Colin Barker_, May 21 2018

%o (GAP) List([2..60], n->3*n^2+38*n-76); # _Muniru A Asiru_, May 20 2018

%Y Cf. A304832.

%K nonn,easy

%O 2,1

%A _Emeric Deutsch_, May 20 2018