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
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).
Programs
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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
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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 *)
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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).
Comments