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a(n) = n^2 + 25*n - 34 (n >=2).
2

%I #20 May 22 2018 08:16:31

%S 20,50,82,116,152,190,230,272,316,362,410,460,512,566,622,680,740,802,

%T 866,932,1000,1070,1142,1216,1292,1370,1450,1532,1616,1702,1790,1880,

%U 1972,2066,2162,2260,2360,2462,2566,2672,2780,2890,3002,3116,3232,3350,3470,3592,3716,3842,3970,4100,4232,4366

%N a(n) = n^2 + 25*n - 34 (n >=2).

%C a(n) is the first Zagreb index of the Mycielskian of the path graph P[n] (n > =2). For the Mycielskian, see p. 205 of the West reference and/or the Wikipedia link.

%C The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums 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="/A304832/b304832.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 a(n) = A132767(n) - 34. - _Felix Fröhlich_, May 20 2018

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

%F G.f.: 2*x^2*(10 - 5*x - 4*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(n^2 + 25*n - 34, n = 2 .. 55);

%o (PARI) a(n) = n^2 + 25*n - 34 \\ _Felix Fröhlich_, May 20 2018

%o (PARI) Vec(2*x^2*(10 - 5*x - 4*x^2) / (1 - x)^3 + O(x^40)) \\ _Colin Barker_, May 21 2018

%o (GAP) List([2..60], n->n^2+25*n-34); # _Muniru A Asiru_, May 20 2018

%Y Cf. A132767, A304833.

%K nonn,easy

%O 2,1

%A _Emeric Deutsch_, May 20 2018