OFFSET
4,1
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.
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).
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).
EXAMPLE
The first Bomfim link shows a way to find a(8).
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 *)
PROG
(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, ", ")))
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Washington Bomfim, Feb 15 2011
STATUS
approved