A186235 Total Wiener index of double-star trees with n nodes.

10, 18, 57, 82, 169, 220, 374, 460, 700, 830, 1175, 1358, 1827, 2072, 2684, 3000, 3774, 4170, 5125, 5610, 6765, 7348, 8722, 9412, 11024, 11830, 13699, 14630, 16775, 17840, 20280, 21488, 24242, 25602, 28689, 30210, 33649, 35340, 39150, 41020

Offset: 4

Author

Washington Bomfim, Feb 15 2011

Keywords

• nonn
• easy

Comments

For the trees of a given order, it appears that the Wiener indexes are very close. For n=8, the indexes are 54, 57, and 58.

The second Bomfim link refers to formulas of the total Wiener index, and the average Wiener index of those trees.

Examples

			The first Bomfim link shows a way to find a(8).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 4..10000
• Rundan Xing, Bo Zho, Ordering trees having small reverse Wiener indices
• W. Bomfim, Example
• W. Bomfim, Formulas
• Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Crossrefs

Cf. A122681, A140106, A168559.

Programs

Magma

[ IsEven(n) select (n-2)*(2*n-3)*(7*n-4)/24 else (n-3)*(n-1)*(7*n-8)/12: n in [4..43] ]; // Bruno Berselli, Feb 17 2011

Mathematica

a[n_]:= a[n] = -a[n-7] + a[n-6] + 3a[n-5] - 3a[n-4] - 3a[n-3] + 3a[n-2] + a[n-1]; a[0]=-1; a[1]=0; a[2]=0; a[3]=0; a[4]=10; a[5]=18; a[6]=57; a /@ Range[4, 43] (* Jean-François Alcover, Jun 01 2011, after recurrence signature *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{10,18,57,82,169,220,374},40] (* Harvey P. Dale, Mar 25 2013 *)

PARI

for(n=4,43,if(n%2,print1((1/12)*(7*n^3+53*n)-3*n^2-2,", "), print1((1/24)*(14*n^3-57*n^2+70*n)-1,", ")))

Formula

G.f.: x^4*(10+8*x+9*x^2+x^3)/((1+x)^3*(1-x)^4). Also a(n)=(n*(28*n^2-129*n+176)+3*(5*n^2-12*n+8)*(-1)^n-72)/48. - Bruno Berselli, Feb 15 2011

For even n, a(n)=(14*n^3-57*n^2+70*n)/24-1, otherwise a(n)=(7*n^3+53*n)/12-3*n^2-2.

With d=floor((n-2)/2), a(n)=d((n-2)*(n-1)+n*(d+3)/2-d^2/3-3*d/2-13/6).