|
| |
|
|
A186235
|
|
Total Wiener index of double-star trees with n nodes.
|
|
1
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
|
|
|
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
|
|
|
|
STATUS
|
approved
|
| |
|
|
