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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
W. Bomfim, Example
W. Bomfim, Formulas
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
Sequence in context: A050576 A144376 A266708 * A241053 A068642 A198309
KEYWORD
nonn,easy
AUTHOR
Washington Bomfim, Feb 15 2011
STATUS
approved

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Last modified July 19 22:34 EDT 2024. Contains 374441 sequences. (Running on oeis4.)