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A186310
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Total Wiener index of star-like trees with n edges.
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0
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1, 4, 19, 54, 145, 350, 726, 1462, 2710, 4846, 8166, 13730, 21741, 34350, 52488, 79518, 117141, 172224, 246993, 353464, 496721, 694952, 958374, 1318174, 1789077, 2420888, 3243045, 4329098, 5728140, 7557906, 9893327, 12913820, 16746136
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OFFSET
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1,2
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COMMENTS
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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:
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)),
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.
n av(n)*A058984(n) (I) a(n) (II) I/II
5 136.9 145 0.94414
13 21443.1 21741 0.98630
20 352132.8 353464 0.99623
28 4329081.3 4329098 0.999996
29 5729910.2 5728140 1.00031
30 7560843.8 7557906 1.00039
33 16760543.2 16746136 1.00086
50 810144542.2 808929430 1.00150
60 5614575632.9 5606027232 1.00152
80 167110984160.2 166870656888 1.00144
100 3203299185861.4 3199052703248 1.00133
120 45208751880788.8 45153537110230 1.00122
130 155331813239050.0 155149438632558 1.00117
140 507674790104504.3 507101038817616 1.00113
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
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REFERENCES
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S. Wagner, Graph-theoretical enumeration and digital expansions: an analytic approach, Dissertation, Fakult. f. Tech. Math. u. Tech. Physik, Tech. Univ. Graz, Austria, Feb. 2006.
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LINKS
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Table of n, a(n) for n=1..33.
Washington Bomfim, Example
Washington Bomfim, Table of n, a(n) for n=1..140
Arnold Knopfmacher, Robert F. Tichy, Stephan Wagner, and Volker Ziegler, Graphs, Partitions and Fibonacci Numbers
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EXAMPLE
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The first Bomfim link shows a way to find a(7).
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CROSSREFS
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Cf. A001620, A002117, A058984, A122681.
Sequence in context: A159833 A166808 A253008 * A122684 A122681 A215862
Adjacent sequences: A186307 A186308 A186309 * A186311 A186312 A186313
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KEYWORD
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nonn
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AUTHOR
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Washington Bomfim, Feb 17 2011
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STATUS
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approved
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