A228318 The Wiener index of the graph obtained by applying Mycielski's construction to the star graph K(1,n).

15, 33, 59, 93, 135, 185, 243, 309, 383, 465, 555, 653, 759, 873, 995, 1125, 1263, 1409, 1563, 1725, 1895, 2073, 2259, 2453, 2655, 2865, 3083, 3309, 3543, 3785, 4035, 4293, 4559, 4833, 5115, 5405, 5703, 6009

Offset: 1

Author

Emeric Deutsch, Aug 27 2013

Keywords

• nonn
• easy

Examples

			a(1)=15; indeed K(1,1) is the 1-edge graph; the Mycielski construction yields the cycle C(5); its Wiener index is 5*1 + 5*2 = 15.
		

References

• D. B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001, p. 205.

Links

• H. P. Patil and R. Pandiya Raj, On the total graph of Mycielski graphs, central graphs and their covering numbers, Discussiones Mathematicae Graph Theory, Vol. 33 (2013), pp. 361-371.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A228319.

Programs

Maple

a := proc (n) options operator, arrow: 4*n^2+6*n+5 end proc; seq(a(n), n = 1 .. 38);

Mathematica

LinearRecurrence[{3,-3,1},{15,33,59},50] (* Harvey P. Dale, Jan 13 2022 *)

PARI

a(n)=4*n^2+6*n+5 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 4*n^2 + 6*n + 5.

G.f.: x*(15-12*x+5*x^2)/(1-x)^3.

The Hosoya-Wiener polynomial is (4*n+1)*t + (2*n^2 + n + 2)*t^2.

From Elmo R. Oliveira, Nov 15 2024: (Start)

E.g.f.: exp(x)*(4*x^2 + 10*x + 5) - 5.

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