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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 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 Cf. A122681, A140106, A168559. Sequence in context: A050576 A144376 A266708 * A241053 A068642 A198309 Adjacent sequences: A186232 A186233 A186234 * A186236 A186237 A186238 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.)