A186310 - OEIS

%I #28 Dec 08 2023 11:39:04

%S 1,4,19,54,145,350,726,1462,2710,4846,8166,13730,21741,34350,52488,

%T 79518,117141,172224,246993,353464,496721,694952,958374,1318174,

%U 1789077,2420888,3243045,4329098,5728140,7557906,9893327,12913820,16746136

%N Total Wiener index of star-like trees with n edges.

%C In the reference, p. 18, theorem 2.14, there is the following formula of the average Wiener index av(n) of a star-like tree with n edges:

%C av(n) = 2*n^2 - (6^(1/2)*n^(3/2))/(2*Pi)*(log(n) + 2*cEuler - log(Pi^2/6) + 24*zeta(3)/(Pi^2)),

%C so an approximate value of a(n) is given by av(n)*A058984(n). The following table was determined approximating zeta(3) by 1.2020569, and Euler's constant by 0.5772156649.

%C     n  av(n)*A058984(n) (I)     a(n) (II)     I/II

%C     5             136.9              145   0.94414

%C    13           21443.1            21741   0.98630

%C    20          352132.8           353464   0.99623

%C    28         4329081.3          4329098   0.999996

%C    29         5729910.2          5728140   1.00031

%C    30         7560843.8          7557906   1.00039

%C    33        16760543.2         16746136   1.00086

%C    50       810144542.2        808929430   1.00150

%C    60      5614575632.9       5606027232   1.00152

%C    80    167110984160.2     166870656888   1.00144

%C   100   3203299185861.4    3199052703248   1.00133

%C   120  45208751880788.8   45153537110230   1.00122

%C   130 155331813239050.0  155149438632558   1.00117

%C   140 507674790104504.3  507101038817616   1.00113

%C For n<=28 the approximation underestimates the actual value of the total Wiener index of star-like trees. For 29 <= n <= 140 it overestimates this total; however as n grows, the rate I/II converges to 1. - _Washington Bomfim_, Feb 17 2011

%H Washington Bomfim, <a href="/A186310/b186310.txt">Table of n, a(n) for n = 1..140</a>

%H Washington Bomfim, <a href="http://oeis.org/wiki/File:Ex.png">Example</a>

%H Arnold Knopfmacher, Robert F. Tichy, Stephan Wagner, and Volker Ziegler, <a href="http://www.math.tugraz.at/~wagner/GraPartFib.pdf">Graphs, Partitions and Fibonacci Numbers</a>

%H Stephan Wagner, <a href="https://math.sun.ac.za/swagner/Diss.pdf">Graph-theoretical enumeration and digital expansions: an analytic approach</a>, Dissertation, Fakult. f. Tech. Math. u. Tech. Physik, Tech. Univ. Graz, Austria, Feb. 2006.

%e The Bomfim link shows a way to find a(7).

%Y Cf. A001620, A002117, A058984, A122681.

%K nonn

%O 1,2

%A _Washington Bomfim_, Feb 17 2011