A304832 a(n) = n^2 + 25*n - 34 (n >=2).

20, 50, 82, 116, 152, 190, 230, 272, 316, 362, 410, 460, 512, 566, 622, 680, 740, 802, 866, 932, 1000, 1070, 1142, 1216, 1292, 1370, 1450, 1532, 1616, 1702, 1790, 1880, 1972, 2066, 2162, 2260, 2360, 2462, 2566, 2672, 2780, 2890, 3002, 3116, 3232, 3350, 3470, 3592, 3716, 3842, 3970, 4100, 4232, 4366

Offset: 2

Author

Emeric Deutsch, May 20 2018

Keywords

• nonn
• easy

Comments

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.

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.

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.

References

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

Links

• Colin Barker, Table of n, a(n) for n = 2..1000
• E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
• Wikipedia, Mycielskian
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A132767, A304833.

Programs

GAP

List([2..60], n->n^2+25*n-34); # Muniru A Asiru, May 20 2018

Maple

seq(n^2 + 25*n - 34, n = 2 .. 55);

PARI

a(n) = n^2 + 25*n - 34 \\ Felix Fröhlich, May 20 2018

PARI

Vec(2*x^2*(10 - 5*x - 4*x^2) / (1 - x)^3 + O(x^40)) \\ Colin Barker, May 21 2018

Formula

a(n) = A132767(n) - 34. - Felix Fröhlich, May 20 2018

From Colin Barker, May 21 2018: (Start)

G.f.: 2*x^2*(10 - 5*x - 4*x^2) / (1 - x)^3.

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

(End)