OFFSET
2,1
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).
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)
MAPLE
seq(n^2 + 25*n - 34, n = 2 .. 55);
PROG
(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
(GAP) List([2..60], n->n^2+25*n-34); # Muniru A Asiru, May 20 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, May 20 2018
STATUS
approved